🧠:[<think>Okay, so I have this problem here: solve the equation 2x² - 3|x| - 2 = 0, find the two real roots α and β, and then compute the value of (αβ)/( |α| + |β| ). The answer choices are given. Let me try to work through this step by step.First, the equation is a quadratic in terms of |x|. Hmm, right. Because |x| is involved, maybe I can substitute y = |x|, which would make the equation 2y² - 3y - 2 = 0. Then solving for y would give me possible values for |x|, and then I can find x from there. That seems like a good plan. Let me try that.So let's set y = |x|. Then substituting into the equation: 2y² - 3y - 2 = 0. Now, solving this quadratic equation. The quadratic formula is y = [3 ± sqrt(9 + 16)] / (4), since the discriminant is b² - 4ac. Here, a = 2, b = -3, c = -2. So discriminant D = (-3)² - 4*2*(-2) = 9 + 16 = 25. So sqrt(25) = 5. Therefore, the solutions are y = [3 + 5]/4 = 8/4 = 2, and y = [3 - 5]/4 = (-2)/4 = -0.5.But wait, y was defined as |x|, which is always non-negative. So y = -0.5 doesn't make sense here. Therefore, the only valid solution is y = 2. So |x| = 2, which means x = 2 or x = -2. Therefore, the roots are 2 and -2. So α and β are 2 and -2. Then, the question is, what is (αβ)/( |α| + |β| )?Wait, but the problem says "Let α, β be the two real roots..." But according to my calculation, the equation only has two roots, 2 and -2. So then α and β are 2 and -2, but since they are roots, maybe the order doesn't matter. So let's compute the numerator first: αβ. That would be 2 * (-2) = -4. Then the denominator: |α| + |β|. Since |2| + |-2| = 2 + 2 = 4. So the value is (-4)/4 = -1. So the answer is -1, which is option A.But wait, hold on a second. The original equation is 2x² - 3|x| - 2 = 0. Let me check if x = 2 and x = -2 are indeed solutions.For x = 2: 2*(2)² - 3*|2| - 2 = 2*4 - 6 - 2 = 8 - 6 - 2 = 0. That works. For x = -2: 2*(-2)² - 3*|-2| - 2 = 2*4 - 6 - 2 = 8 - 6 - 2 = 0. That also works. So those are valid roots.But wait, I solved for y = |x|, got y = 2 and y = -0.5. Discarded y = -0.5 since |x| can't be negative, so only y = 2. So x = ±2. Therefore, only two roots. So the problem says "the two real roots α, β". Then the roots are 2 and -2. Therefore, the product αβ is -4, and |α| + |β| is 4, so their ratio is -1. So answer is A.But wait, the answer choices include -1 as option A, so that's possible. But let me check if there's any mistake here because sometimes when dealing with absolute values, especially in quadratic equations, there might be other roots or considerations.Wait, the original equation is 2x² - 3|x| - 2 = 0. Let me consider the cases when x ≥ 0 and x < 0 separately, just to make sure.Case 1: x ≥ 0. Then |x| = x, so the equation becomes 2x² - 3x - 2 = 0. Solving this quadratic equation: discriminant D = 9 + 16 = 25. Solutions are [3 ± 5]/4. So (3 + 5)/4 = 2, and (3 - 5)/4 = -0.5. But since x ≥ 0, x = 2 is valid, x = -0.5 is invalid. So only x = 2 in this case.Case 2: x < 0. Then |x| = -x, so the equation becomes 2x² - 3(-x) - 2 = 0 → 2x² + 3x - 2 = 0. Solving this quadratic: discriminant D = 9 + 16 = 25. Solutions are [-3 ± 5]/4. So (-3 + 5)/4 = 0.5, (-3 -5)/4 = -2. Since x < 0, x = 0.5 is invalid, x = -2 is valid. So only x = -2.Therefore, the roots are indeed 2 and -2. So that's correct. So α and β are 2 and -2. Then, as before, αβ = -4, |α| + |β| = 4, so the ratio is -1. So answer A. So that should be correct.But let me check the answer options again. The options are (A) -1, (B) 1, (C) -2/3, (D) 2/3. So A is -1. So seems correct. But let me see if the problem is trying to trick me in any way.Wait, the problem says "the two real roots of the equation 2x² - 3|x| - 2 = 0". So since the equation is quadratic in |x|, and the solutions for |x| are 2 and -0.5, but we discard the negative one, so |x| = 2, leading to x = 2 and x = -2. Therefore, two real roots. So yes, that's correct. Then compute (αβ)/( |α| + |β| ). So that's (-4)/(2 + 2) = -4/4 = -1. So answer is A. So seems correct.Wait, but maybe the problem is presented as a quadratic in x, but due to the absolute value, the equation is not a standard quadratic. However, splitting into cases as I did gives the same result. So seems correct. So I think the answer should be A, -1.But just to make sure, let's check once again. If α = 2 and β = -2, then product is -4. |α| is 2, |β| is 2, sum is 4. So -4 /4 = -1. So yeah. Therefore, answer is A. Therefore, I think that's correct.Final Answerboxed{A}</think>Given the equation (2x^2 - 3|x| - 2 = 0), we need to find the two real roots (alpha) and (beta) and compute the value of (frac{alpha beta}{|alpha| + |beta|}).First, we substitute (y = |x|), transforming the equation into a quadratic equation in (y):[2y^2 - 3y - 2 = 0]Solving this quadratic equation using the quadratic formula (y = frac{-b pm sqrt{b^2 - 4ac}}{2a}) where (a = 2), (b = -3), and (c = -2):[y = frac{3 pm sqrt{9 + 16}}{4} = frac{3 pm 5}{4}]This gives the solutions:[y = frac{8}{4} = 2 quad text{and} quad y = frac{-2}{4} = -0.5]Since (y = |x|) must be non-negative, we discard (y = -0.5). Thus, (y = 2) implies (|x| = 2), leading to (x = 2) or (x = -2). Therefore, the roots (alpha) and (beta) are 2 and -2.Next, we compute the required value:[frac{alpha beta}{|alpha| + |beta|} = frac{(2)(-2)}{|2| + |-2|} = frac{-4}{2 + 2} = frac{-4}{4} = -1]Thus, the value is (boxed{A}).,<think>Okay, let me try to work through this problem step by step. So, the equation given is 2x² - 3|x| - 2 = 0, and we need to find the value of αβ/(|α| + |β|) where α and β are the two real roots. The options are A to D, with D being 2/3, which might be a hint, but I need to verify.First, I remember that equations involving absolute values can sometimes be split into cases depending on whether the variable inside the absolute value is positive or negative. Since the absolute value is around x here, maybe I can consider two cases: when x ≥ 0 and when x < 0. That way, I can remove the absolute value and solve the quadratic equation in each case.Let me start with the case when x ≥ 0. In that case, |x| = x, so the equation becomes 2x² - 3x - 2 = 0. Hmm, that's a standard quadratic equation. Let me try to solve that. The quadratic formula is x = [3 ± sqrt(9 + 16)] / 4, because the discriminant is b² - 4ac = 9 + 16 = 25. So sqrt(25) is 5. Therefore, the solutions are (3 + 5)/4 = 8/4 = 2 and (3 - 5)/4 = (-2)/4 = -0.5. But wait, in this case, we assumed x ≥ 0, so the negative solution x = -0.5 is not valid here. Therefore, in the x ≥ 0 case, the only solution is x = 2.Now, moving on to the case when x < 0. In that case, |x| = -x, because x is negative. So substituting into the original equation, we get 2x² - 3(-x) - 2 = 0, which simplifies to 2x² + 3x - 2 = 0. Let me solve this quadratic equation as well. Using the quadratic formula again, x = [-3 ± sqrt(9 + 16)] / 4. Wait, the discriminant here is the same: b² - 4ac = 9 + 16 = 25. So sqrt(25) is 5. Therefore, the solutions are (-3 + 5)/4 = 2/4 = 0.5 and (-3 -5)/4 = -8/4 = -2. However, in this case, we assumed x < 0, so x = 0.5 is not valid here. Therefore, the only solution in this case is x = -2.So the two real roots of the equation are x = 2 and x = -2. Therefore, α and β are 2 and -2. Now, we need to compute αβ/(|α| + |β|). Let me compute each part:First, αβ is 2 * (-2) = -4.Next, |α| is |2| = 2, and |β| is |-2| = 2. So |α| + |β| is 2 + 2 = 4.Therefore, the value is -4 / 4 = -1. Wait, that's answer choice A. But looking at the options, A is -1, B is 1, C is -2/3, D is 2/3. Hmm, but the answer I got is -1, which is option A. However, I need to check if I made a mistake here.Wait, the problem says "the two real roots". Let me check my solutions again. For the case x ≥ 0, equation becomes 2x² -3x -2 =0. Solving, x = [3 ±5]/4. So 2 and -0.5. But since x ≥0, we take 2. For x <0, equation becomes 2x² +3x -2=0. Solutions are [ -3 ±5 ]/4, so 0.5 and -2. Since x <0, we take -2. So the roots are 2 and -2. Therefore, α and β are 2 and -2. Then, αβ is (2)(-2) = -4. |α| is 2, |β| is 2. So |α| + |β| is 4. Therefore, -4 /4 is -1. So the answer is A? But the options given include A as -1. But I thought maybe D is the correct answer. Wait, maybe I did something wrong here.Wait, hold on. Let me check the original equation again. The equation is 2x² -3|x| -2=0. Let me verify if x = 2 is a solution. Plugging in: 2*(4) -3*(2) -2 = 8 -6 -2 =0. Yes, that works. For x = -2: 2*(4) -3*(2) -2 = same thing, 8 -6 -2=0. So yes, both roots are valid. So α and β are 2 and -2. Then αβ is -4, and |α| + |β| is 4, so the value is -1. But option A is -1, which is present. However, maybe there is a mistake here. Wait, but the problem says "the two real roots". Wait, but maybe I missed something. Let me think again.Wait, in the case when x >=0, the equation is 2x² -3x -2=0, which has roots 2 and -0.5. But we discard -0.5. Then in x <0, the equation is 2x² +3x -2=0, roots 0.5 and -2. Discard 0.5. So the only roots are 2 and -2. So that's correct.Alternatively, maybe the problem is considering that when solving for x, even if x is replaced by |x|, but no, the original equation is in x, not |x|. So the roots are indeed 2 and -2. Then, the value is -1. So answer A.But the options include D as 2/3, which is also an option. Let me check again. Maybe there's a mistake in calculation.Wait, αβ is 2*(-2) = -4. |α| + |β| is 2 + 2 =4. So -4/4 is -1. That's correct. Therefore, answer is A. But I need to check the problem again.Wait, the problem says "Let α, β be the two real roots of the equation 2x² -3|x| -2=0. Then the value of αβ/(|α| + |β|) is..." So if α and β are 2 and -2, then yes, the answer is -1. However, maybe there's a different way of interpreting the problem.Wait, perhaps the problem is written as 2x² -3|x| -2 =0. Maybe when we split into cases, we have to solve for |x| first. Let's see. Let me try substituting y = |x|. Then, the equation becomes 2y² -3y -2=0. Then solving for y. So this quadratic equation is 2y² -3y -2=0. Which is similar to the first case. Let's solve this equation for y.Using quadratic formula: y = [3 ± sqrt(9 + 16)] /4 = [3 ±5]/4. So y = (3 +5)/4=8/4=2 or y=(3-5)/4=-2/4=-0.5. But since y = |x|, y cannot be negative. Therefore, the only solution is y=2. So |x|=2, which implies x=2 or x=-2. Therefore, the roots are 2 and -2. So same as before.Therefore, α and β are 2 and -2. Then, αβ = -4, |α| + |β| =4, so ratio is -1. So answer is A. But in the options given, A is -1. So that's correct.Wait, but the answer in the options is (D) 2/3. Wait, maybe I made a mistake here.Wait, no. Wait, let me check again. If the roots are 2 and -2, then αβ is -4. |α| + |β| is 2 + 2=4. So -4/4 is -1. So answer is A. But maybe the question is different? Wait, let me check the problem again. It says "the value of αβ/(|α| + |β|)". Yes. So maybe the answer is A. But I need to check if I have the right roots.Alternatively, perhaps there's a mistake in the problem statement? Let me check.The equation is 2x² -3|x| -2=0. If x is positive, |x|=x, so 2x² -3x -2=0. Solutions x=2 and x=-0.5 (discarded). If x is negative, |x|=-x, so 2x² +3x -2=0. Solutions x=0.5 (discarded) and x=-2. So the roots are 2 and -2. Therefore, αβ is -4. |α| + |β|=4. So ratio is -1. So answer is A. Therefore, the correct answer should be A. But maybe I made a mistake in calculation.Wait, but let me check with another approach. Let's substitute y = |x|, so equation becomes 2y² -3y -2=0. As before, y=2. So |x|=2, x=±2. Therefore, roots are 2 and -2. Then the same result.Alternatively, maybe the question is expecting us to take α and β as the roots of the equation in terms of y, which would be y=2, but since y=|x|, so x is ±2. But maybe if we consider the equation as in terms of y, the product of roots (in terms of y) is -2/2=-1. But then, but here, the roots are 2 and -2, which are x's. So maybe the product of the x roots is -4. So, but in terms of y, the equation is 2y² -3y -2=0, product of roots is c/a = -2/2=-1. But that's the product of the y roots. But y=|x|, so the product of |x| roots would be -1. But since |x| is always positive, but the product is -1, which is negative. That seems contradictory. Wait, no, actually, in the equation 2y² -3y -2=0, the roots are y=2 and y=-0.5, but since y cannot be negative, we discard y=-0.5, so only y=2. Therefore, there's only one root for y, which is 2. Therefore, |x|=2, so x=2 and x=-2. Therefore, the x roots are 2 and -2.So in the original equation, the quadratic in x is not a standard quadratic because of the absolute value, but splitting into cases gives two roots. Therefore, the product of the roots (in x) is (2)*(-2)= -4. The sum of |α| + |β| is 2 + 2=4. So -4/4= -1.Therefore, answer is A. But the options given include A as -1, so that's the correct answer.Wait, but I just realized, maybe the problem is written differently. Let me check again:"Let α, β be the two real roots of the equation 2x² -3|x| -2=0. Then the value of αβ/(|α| + |β|) is ( )."Yes, exactly. So, with that, the answer is A. However, in some textbooks, sometimes there's confusion between roots and solutions, but in this case, the solutions are x=2 and x=-2, so they are the roots.But wait, let me check if the problem is maybe written as 2x² -3|x| -2=0. If I graph this equation, it should be symmetric about the y-axis because of the absolute value. So the graph would have a V-shape. Let me confirm by plotting or thinking about the graph.For x ≥0, the equation is 2x² -3x -2. The graph here is a parabola opening upwards, with vertex at x=3/(4), which is positive. At x=0, the value is -2. Then, it crosses the x-axis at x=2. For x <0, the equation becomes 2x² +3x -2. This is also a parabola opening upwards, with vertex at x=-3/(4). At x=0, it's -2. It crosses the x-axis at x=-2. So the graph crosses the x-axis at x=2 and x=-2, so those are the only real roots. Therefore, confirming again that the roots are 2 and -2.Therefore, the calculation is correct. Therefore, the answer is -1, which is option A. But the initial thought was maybe D, but that's incorrect. Therefore, the correct answer is A.Wait, but the problem is given in a multiple-choice question where options are A to D. Let me check again the options:(A) -1(B) 1(C) -2/3(D) 2/3So A is -1, which matches our result. So why did I initially think D? Maybe because of a miscalculation, but after checking step by step, it's clear that A is correct. Therefore, the answer is A.However, just to ensure, let me check once more.Compute αβ/(|α| + |β|):α = 2, β = -2.αβ = (2)(-2) = -4.|α| = 2, |β| = 2, so |α| + |β| = 4.Thus, -4 / 4 = -1. Yes, so answer is A. Therefore, the correct option is A.But wait, hold on. The problem says "the two real roots". Wait, but is the equation 2x² -3|x| -2=0 a quadratic equation? Actually, because of the absolute value, it's not a quadratic equation in x, but piecewise quadratic. However, by substituting y = |x|, we get a quadratic in y, but only one valid solution y=2, leading to two solutions in x. So in terms of x, there are two real roots: 2 and -2.Therefore, everything checks out. So I think the answer is indeed A.Final Answerboxed{A}</think>Given the equation (2x^2 - 3|x| - 2 = 0), we need to find the value of (frac{alpha beta}{|alpha| + |beta|}) where (alpha) and (beta) are the two real roots.First, consider the case when (x geq 0). The equation becomes (2x^2 - 3x - 2 = 0). Solving this quadratic equation using the quadratic formula:[x = frac{3 pm sqrt{9 + 16}}{4} = frac{3 pm 5}{4}]This gives the roots (x = 2) and (x = -0.5). Since (x geq 0), we discard (x = -0.5), leaving (x = 2).Next, consider the case when (x < 0). The equation becomes (2x^2 + 3x - 2 = 0). Solving this quadratic equation using the quadratic formula:[x = frac{-3 pm sqrt{9 + 16}}{4} = frac{-3 pm 5}{4}]This gives the roots (x = 0.5) and (x = -2). Since (x < 0), we discard (x = 0.5), leaving (x = -2).Thus, the two real roots are (alpha = 2) and (beta = -2).Calculate (alpha beta):[alpha beta = 2 times (-2) = -4]Calculate (|alpha| + |beta|):[|alpha| + |beta| = |2| + |-2| = 2 + 2 = 4]Finally, the value of (frac{alpha beta}{|alpha| + |beta|}) is:[frac{-4}{4} = -1]Thus, the answer is (boxed{A}).]

🔑:2. A.The original equation is (2|x|+1)(|x|-2)=0.Since 2|x|+1>0,|x|-2=0, therefore, x= pm 2. Hence, frac{a beta}{|alpha|+|beta|}=frac{2 times(-2)}{|2|+|-2|}=-1.

🧠:[<think>Okay, so I have this problem where there's a prime number p. On the left side of the board, numbers from 1 to p-1 are written, and on the right side, there's a 0. We perform p-1 moves. Each move involves selecting a number from the left side, adding it to all the remaining numbers on the board (both left and right sides), and then erasing the selected number. The goal is to determine for which primes p it's possible to choose the numbers in such a way that the final remaining number on the board is divisible by p.Hmm, let me start by understanding the process step by step. Maybe working through a small prime example would help. Let's take p=2 first. Then the left side has 1, and the right side has 0. Since p-1=1, we only need to do one move: select 1, add it to all remaining numbers. But after selecting 1, there's only 0 left. Adding 1 to 0 gives 1. So the final number is 1, which is not divisible by 2. So for p=2, it's not possible? Wait, but maybe I did that wrong. Let me check again.Wait, when p=2, left side is [1], right side is [0]. The move is to select 1, add it to all remaining numbers. The remaining numbers after selecting 1 would be just 0. Adding 1 to 0 gives 1. So yes, the final number is 1, which mod 2 is 1, so not divisible. So p=2 is not possible.Now p=3. Left side: 1, 2; right side: 0. We need to perform 2 moves. Let's see. Let's try the first move: pick 1. Then add 1 to the remaining numbers. The remaining numbers on the left would be 2, and the right is 0. So after adding 1, they become 2+1=3 and 0+1=1. Then erase the selected number 1. So now left side has 3, but since p=3, we might need to consider modulo 3? Wait, does the problem say to reduce modulo p at each step? The problem statement says "add it to all the remaining numbers on the board", but it doesn't specify modulo. But since we want the final number to be divisible by p, maybe all operations are done in integers, and then the final number is checked for divisibility by p. Hmm. So in the case of p=3, after first move: left side becomes 3 (but 3 is equal to p, so if we erase it? Wait, no, 3 is on the left side? Wait, originally, left side is 1,2 and right is 0. After picking 1, we add 1 to remaining left number 2 and right number 0. So left becomes 3 and right becomes 1. But 3 is now on the left? Or does the left side only contain numbers from 1 to p-1, and once you add, they can go beyond? Hmm, the problem statement doesn't restrict the numbers to be within 0 to p-1. So we just have numbers as integers, and the final number has to be divisible by p.But then when we do the next move, the left side has 3 and right has 1. Then the second move is to pick a number from the left side, which is 3. Then we add 3 to all remaining numbers. The remaining numbers are the right side 1. So adding 3 to 1 gives 4. Then erase 3. So the final number is 4, which modulo 3 is 1. So not divisible by 3. So that sequence doesn't work.Alternatively, what if in the first move we pick 2 instead of 1. Then add 2 to the remaining left number 1 and right number 0. So left becomes 1+2=3, right becomes 0+2=2. Then erase 2. Then the left side is 3, right is 2. Then the second move: pick 3 from the left. Add 3 to remaining right number 2. So 2+3=5. Erase 3. Final number is 5, which modulo 3 is 2. Not divisible. So p=3 also doesn't work? But maybe there's another way. Wait, but after first move, left side numbers can be larger than p. So maybe we need to consider modulo operations during the process? The problem didn't specify, but the final number just needs to be divisible by p. So maybe even if the intermediate numbers are large, as long as the final number is divisible by p. Let me check.Wait, in the first approach for p=3, when we picked 1 first, then 3, we ended up with 4. 4 mod 3 is 1. If we had picked 2 first, then 3, ended up with 5 mod 3 is 2. So neither is 0. So maybe p=3 is impossible. Hmm.Wait, but maybe I made a mistake in the process. Let me re-examine the problem statement.Each move: select one of the numbers on the left side, add it to all remaining numbers on the board, then erase the selected number.So in the first step, when we select a number from the left, we add it to the remaining numbers on the left and the right. So, for example, in p=3: left is 1,2; right is 0. If I pick 1, then I add 1 to 2 (remaining left) and 0 (right). So left becomes 3, right becomes 1. Then erase 1. Then next move: left is 3, right is 1. Now, left side only has 3. So pick 3, add 3 to remaining number 1. Result is 4. So final number 4. Not divisible by 3.Alternatively, if in the first move, pick 2: add 2 to 1 and 0. Left becomes 3, right becomes 2. Then pick 3, add 3 to 2: 5. Not divisible.So p=3 seems impossible. What about p=5? Let's try p=5. Maybe trying a larger prime can reveal a pattern.Left side: 1,2,3,4; right side: 0. We need to perform 4 moves.This might get complicated, but let's try to find a sequence where the final number is 0 mod 5.Alternatively, maybe there's a pattern here. Let's think algebraically. Let me try to model the operations.At each step, when we select a number x from the left, we add x to all remaining numbers on the board (both left and right). Then we erase x. So each remaining number is increased by x. So the key is that each number that remains after x is selected is increased by x. Then x is removed.So perhaps we can model the effect of each operation on the numbers. Let's denote the numbers on the left as a set S, and the right as a single number R (initially 0). Each move, pick an element x from S, add x to all elements in S{x} and to R, then remove x from S.So after each move, the new set S' = {y + x | y ∈ S {x}} and R' = R + x.But since we're dealing with primes and divisibility, maybe we can consider all operations modulo p. Since we're interested in the final number being 0 mod p, perhaps we can track the numbers modulo p throughout the process. That might simplify things.So let me consider all operations modulo p. Then, starting with S = {1,2,...,p-1} and R = 0.Each move: pick x from S. Then S becomes { (y + x) mod p | y ∈ S {x} }, and R becomes (R + x) mod p.Wait, but actually, adding x to all remaining numbers is equivalent to adding x to each remaining number. So modulo p, each of these additions is modulo p.But perhaps instead of working modulo p, we can think in terms of linear algebra or permutations. Wait, each time we pick a number, we add it to the others. This is similar to elementary row operations in linear algebra, where adding a multiple of one row to another. But here, it's adding the selected number to all remaining numbers, not just one.Alternatively, since each operation affects all remaining numbers by adding x, which is the number being removed. Let's think of the total effect after all operations.Suppose we have a sequence of numbers x_1, x_2, ..., x_{p-1} being removed in each step. Then, each remaining number after each step is added by x_i. So the order in which we remove the numbers affects how many times each x_i is added to the remaining numbers.Wait, for example, the first number removed is x_1. It gets added to all the other p-2 numbers on the left and to the right (which is 0). Then the second number removed is x_2. At that point, there are p-2 numbers left on the left. So x_2 is added to the remaining p-3 numbers on the left and to the current right number (which is x_1). And so on, until the last number removed is x_{p-1}, which is added only to the right number.So, the final right number R will be the sum of all the x_i's. Because each x_i is added to R exactly once when it's removed. But also, each time a number is removed, all the other numbers (left and right) are increased by x_i. So the numbers on the left are being increased by all x_j's that are removed after they themselves are removed. Wait, no, actually, when you remove x_i, you add x_i to all remaining numbers (left and right). So the remaining left numbers after x_i is removed will have been increased by x_i. Similarly, the right number is increased by x_i each time a left number is removed.But the right number starts at 0 and each x_i is added to it exactly once when x_i is removed. Wait, no: each time you remove a left number, you add it to the right number. So if you remove x_1 first, the right number becomes 0 + x_1. Then, when you remove x_2, the right number becomes x_1 + x_2. Continuing, after all p-1 moves, the right number is x_1 + x_2 + ... + x_{p-1}.But also, the left numbers are being modified. However, the last remaining left number is modified by all the previous additions. Wait, the final remaining left number is... Hmm, no, after p-1 moves, all left numbers are removed, right? Wait, initially there are p-1 numbers on the left. Each move removes one, so after p-1 moves, there are no numbers left on the left. The right number is the only one remaining. So the final number is the sum of all x_i's. Wait, that can't be. Because when you remove x_i, you add it to all remaining left numbers and the right number. So the right number accumulates the sum of all x_i's. But the left numbers also get incremented by each x_j where j is before they are removed.Wait, let's try to track the right number. Each time a left number is removed, the right number is increased by x_i. So the right number ends up being x_1 + x_2 + ... + x_{p-1}. So regardless of the order in which we remove the numbers, the final right number is the sum of all left numbers. But the left numbers are 1, 2, ..., p-1. So the sum is (p-1)p/2. Therefore, the final number is (p-1)p/2. Then, we need this to be divisible by p. So (p-1)p/2 ≡ 0 mod p. Since p is prime, p divides (p-1)p/2. Since p divides the term, then (p-1)/2 must be an integer? Wait, no. Wait, (p-1)p/2 ≡ 0 mod p is equivalent to p dividing (p-1)p/2. Since p divides p, the entire term is divisible by p, but we have (p-1)/2 times p. So (p-1)/2 doesn't have to be an integer. Wait, but (p-1)p/2 is divisible by p if and only if (p-1)/2 is an integer, which would require that p-1 is even, so p is odd. But all primes except 2 are odd. So for p=2, (2-1)*2/2 = 1, which is not divisible by 2. For odd primes, (p-1) is even, so (p-1)/2 is integer. Therefore, (p-1)p/2 is divisible by p if and only if (p-1)/2 is divisible by 1 (since p and 2 are coprime). Wait, no. Wait, (p-1)p/2 divided by p is (p-1)/2, so (p-1)p/2 ≡ 0 mod p if and only if (p-1)/2 ≡ 0 mod 1, which is always true. Wait, that can't be.Wait, no. Let me think again. If we have (p-1)p/2 ≡ 0 mod p, then this is equivalent to p dividing (p-1)p/2. But since p divides (p-1)p/2, and p is prime, then p divides either (p-1) or p/2. But p divides p/2 only if 2 divides p, which would mean p=2. For p>2, since p is prime and greater than 2, it's odd, so 2 doesn't divide p. Therefore, p divides (p-1) only if p divides (p-1), which is impossible because p > p-1. Therefore, only when p divides p/2, which only happens when p=2. But for p=2, (2-1)*2/2=1, which is not 0 mod 2. Therefore, (p-1)p/2 is never divisible by p. Wait, this contradicts the initial logic. Where did I go wrong?Wait, maybe my assumption that the final number is the sum of all x_i's is incorrect. Wait, perhaps the right number is not just the sum of x_i's, because each time we remove a left number, we also add x_i to the remaining left numbers. So the left numbers are being modified, which in turn affect the subsequent additions. Therefore, my previous conclusion that the final right number is simply the sum of x_i's is wrong. Therefore, I need to model this more carefully.Let me try to model the process mathematically. Let's denote the numbers on the left as a multiset L and the right number as R. Initially, L = {1, 2, ..., p-1} and R = 0.Each move, we pick a number x from L, add x to all elements in L {x} and to R, then remove x from L.Therefore, after the first move:L becomes {y + x | y ∈ L {x}}.R becomes R + x.But note that when we add x to the remaining elements, those elements are then part of the new L for the next move.This process is similar to performing operations where each remaining number is augmented by the sum of the numbers removed before them. Let me see.Suppose we remove x_1 first. Then all remaining numbers in L are increased by x_1, and R becomes x_1.Then, in the next move, we remove x_2 (which was originally some number y_2, but after the first addition, it's y_2 + x_1). Then, adding x_2 to the remaining numbers (which are now (L {x_1, y_2}) + x_1) would result in those numbers being increased by x_2, and R becomes x_1 + x_2.Wait, this seems complicated. Maybe instead, we can model each number's contribution to the final R.Alternatively, let's track the value of R through each step. Initially, R = 0.Each time we remove a number x_i, R increases by x_i. But also, each time we remove a number x_i, all the remaining numbers in L (including those that will be removed later) are increased by x_i. Therefore, when we later remove another number x_j, that x_j has already been increased by all x_i's that were removed before it.Therefore, the final R is the sum of all x_i's, where each x_i is the value of the number at the time it was removed. But since each x_i is added to all subsequent numbers, the order matters.This seems like each removed number x_k is equal to the original number plus the sum of all x_i's removed before x_k.Wait, let me try with p=3. Let's take p=3 again.Original L: {1, 2}, R=0.Case 1: Remove 1 first.- Add 1 to remaining L (which is 2) and R.- New L: {2 + 1} = {3}- New R: 0 + 1 = 1Then remove 3:- Add 3 to remaining L (none) and R.- New R: 1 + 3 = 4Final R = 4 ≡ 1 mod 3.Case 2: Remove 2 first.- Add 2 to remaining L (1) and R.- New L: {1 + 2} = {3}- New R: 0 + 2 = 2Then remove 3:- Add 3 to R.- New R: 2 + 3 = 5 ≡ 2 mod 3.So regardless of the order, R ends up as 1 or 2 mod 3, neither 0. So p=3 doesn't work. Hmm.Wait, but maybe there's another way? But with p=3, there are only two possible orders: remove 1 first or remove 2 first. Both result in non-zero. So p=3 is impossible.Now let's try p=5. This might be tedious, but let's attempt it.Original L: {1,2,3,4}, R=0.Suppose we remove numbers in reverse order: 4,3,2,1.First move: Remove 4.- Add 4 to remaining L {1,2,3} and R.- New L: {1+4=5, 2+4=6, 3+4=7} = {5,6,7}- But since we can ignore the actual values and consider modulo 5, these are equivalent to 0,1,2.Wait, but perhaps we should keep track of actual values, not modulo. Because the problem requires the final number to be divisible by p, not necessarily that intermediate steps are modulo p. So let's keep actual values.After removing 4:L: {5,6,7}, R: 0 + 4 = 4.Second move: Remove 3 (original 3, but now it's 7? Wait, hold on. Wait, after removing 4, the remaining numbers in L are 1,2,3, each increased by 4. So they become 5,6,7. Then we remove one of these. Wait, but the problem states we are to remove numbers from the left side, which initially are 1,2,3,4. But after the first move, the left side is {5,6,7}. So do we consider these as new numbers, or do we still refer to the original numbers? Wait, the problem says "select one of the numbers written on the left side of the board", so after the first move, the left side has {5,6,7}, so we can select any of these.But if we are trying to remove in reverse order, maybe we need to adjust. Wait, perhaps this approach is too confusing. Let's instead try to find an invariant or mathematical formula for the final R.Alternatively, think of each number's contribution to R. Each time a number is removed, it is added to R. Additionally, every subsequent removal will add the current value of the number being removed, which has been augmented by previous additions.This seems recursive. Let me see if there's a pattern.Suppose we have numbers a_1, a_2, ..., a_{p-1} on the left. Each time we remove a number, say a_i, we add a_i to all remaining numbers and to R. Then, the next numbers will be a_j + a_i for each remaining a_j, and R becomes R + a_i.If we continue this process, the final R will be the sum of all the a_i's, each multiplied by 2^{k_i}, where k_i is the number of times it's been added to subsequent numbers. Wait, perhaps not exactly. Let's model it.Alternatively, let's model the coefficient of each original number in the final R.Each original number x in L will contribute to R in two ways: once when it's removed (added to R), and also, every time a number is removed before it, x is added to the subsequent numbers, which might then be removed later, contributing their own values (which include x).This seems complex. Let me consider the contribution of each original number to the final R.Suppose the order of removal is x_1, x_2, ..., x_{p-1}. Then:- When x_1 is removed, it is added to R and to all other numbers. So R gets x_1, and each remaining x_i (i >1) becomes x_i + x_1.- When x_2 is removed, it is the value x_2 + x_1 (since x_1 was added to it). Then, R gets x_2 + x_1, and each remaining x_j (j >2) becomes (x_j + x_1) + (x_2 + x_1) = x_j + x_1 + x_2 + x_1 = x_j + 2x_1 + x_2. Wait, no, that's incorrect. Wait, when we remove x_2, we add x_2 (which is already x_2 + x_1) to the remaining numbers. So remaining numbers after x_1 is removed are x_2 + x_1, x_3 + x_1, etc. When we remove x_2 + x_1, we add that to the remaining numbers. So the new remaining numbers would be (x_3 + x_1) + (x_2 + x_1) = x_3 + 2x_1 + x_2.Wait, perhaps this approach is getting too convoluted. Let's think recursively. Let me denote that each number removed contributes its current value to R, and all subsequent numbers are increased by that value. Therefore, the contribution of each original number to R can be represented as the sum of itself plus the sum of all numbers removed before it.Wait, here's a different approach. Suppose we have numbers 1, 2, ..., p-1. Let's consider the order in which we remove them. Let's say we remove them in the order π(1), π(2), ..., π(p-1), where π is a permutation of 1 to p-1. Then, the final R is equal to the sum over i=1 to p-1 of (π(i) multiplied by 2^{i-1}). Wait, is that true? Wait, no, but maybe each time you remove a number, its contribution is doubled for the next steps. Let me think.Alternatively, when you remove the first number, x_1, it's added to R once and to all other numbers. Then, when you remove the second number, x_2, which is now x_2 + x_1, you add that to R, so R gets x_1 + (x_2 + x_1) = 2x_1 + x_2. Then, when you remove the third number, which is x_3 + x_1 + (x_2 + x_1) = x_3 + 2x_1 + x_2, so R becomes 2x_1 + x_2 + (x_3 + 2x_1 + x_2) = 4x_1 + 2x_2 + x_3. This seems to follow a pattern where each term is multiplied by 2 each step. So perhaps the coefficient of x_k is 2^{p-1 - k}?Wait, maybe the total R is the sum over each x_i multiplied by 2^{number of removals after x_i}. Wait, if we remove x_i first, it affects all subsequent removals. So each x_i's contribution is x_i multiplied by 2^{n - i}, where n is p-1. Wait, maybe similar to binary representations. Let's test with p=3.If we remove x_1 first, then x_2. Then R = x_1 + (x_2 + x_1) = 2x_1 + x_2. If we remove x_2 first, then x_1. Then R = x_2 + (x_1 + x_2) = x_2 + x_1 + x_2 = x_1 + 2x_2. So for p=3, the final R is either 2x_1 + x_2 or x_1 + 2x_2, depending on the order.Given that x_1 and x_2 are 1 and 2 (for p=3). If we remove 1 first, R = 2*1 + 2 = 4. If we remove 2 first, R = 1 + 2*2 = 5. Both 4 and 5 modulo 3 are 1 and 2. So indeed, neither is 0. So p=3 is impossible.Wait, but if the final R is a linear combination of the original numbers with coefficients that are powers of 2, then for R to be 0 mod p, the combination must be 0 mod p. So this is equivalent to solving 2^{p-2}x_1 + 2^{p-3}x_2 + ... + 2^{0}x_{p-1} ≡ 0 mod p, where x_i are the numbers 1, 2, ..., p-1 in some order.But since the permutation can be any order, maybe there exists a permutation where this linear combination is 0 mod p. So the problem reduces to whether there's a permutation of 1, 2, ..., p-1 such that the sum with coefficients 2^{p-2}, 2^{p-3}, ..., 1 is divisible by p.Alternatively, note that the coefficients are powers of 2. Let's denote the coefficients as c_i = 2^{p-1 - i} for i from 1 to p-1. Then the sum is c_1x_1 + c_2x_2 + ... + c_{p-1}x_{p-1}.But since the permutation is arbitrary, we can rearrange the x_i's. So we need to find if the multiset {1, 2, ..., p-1} can be permuted such that the weighted sum with weights c_i is 0 mod p.But since the weights are fixed (only depending on the position), and we can permute the numbers, this is equivalent to whether the set {c_1, c_2, ..., c_{p-1}} is a complete set of residues mod p, or if there exists a permutation such that the sum matches.Alternatively, maybe there's a relation between the weights and the multiplicative inverses. Alternatively, since the weights are geometric series, maybe we can sum them up.Wait, for each permutation, the sum is a different linear combination. However, since we can permute the numbers, we are essentially looking for a rearrangement such that the dot product of the weight vector and the permuted number vector is 0 mod p.This seems related to the concept of orthogonality in vector spaces over finite fields. But maybe there's a simpler way.Let me consider the sum of all possible permutations. Since the numbers 1 to p-1 are a complete residue system mod p (excluding 0), and the weights are fixed coefficients. But this might not help.Alternatively, notice that the weights form a geometric progression. The coefficients are 2^{p-2}, 2^{p-3}, ..., 2^0. So the sum of the weights is 2^{p-1} - 1. Wait, no, the sum of a geometric series from 2^0 to 2^{n-1} is 2^n - 1. Here, the weights are from 2^{0} to 2^{p-2}, so the sum is 2^{p-1} - 1.But how does this relate?Alternatively, maybe consider that for the final R to be 0 mod p, the weighted sum must be 0 mod p. Since the weights are c_i = 2^{p-1 - i}, and the numbers are 1 to p-1 in some order.If we consider all permutations, the total number of different sums modulo p is probably quite large. So it's possible that for some primes, such a permutation exists, and for others, not.Alternatively, maybe there's a connection to the multiplicative order of 2 modulo p. For example, if 2 is a primitive root modulo p, or if the order of 2 divides (p-1), which it always does by Fermat's little theorem.Alternatively, since the weights are 2^{p-2}, 2^{p-3}, ..., 2^0. Let's factor out 2^{0} from the last term, 2^{1} from the second last, etc. Wait, maybe writing the sum as:sum_{k=0}^{p-2} 2^{k} * x_{p-1 -k}where x_{p-1 -k} is a permutation of 1 to p-1.So we need sum_{k=0}^{p-2} 2^{k} * x_{k} ≡ 0 mod p, where x_k is a permutation of 1 to p-1.This resembles a linear combination where coefficients are powers of 2 and variables are the numbers 1 to p-1.Since the permutation is arbitrary, we can assign each x_k to any of the numbers 1 to p-1. So the question is whether the set {2^0, 2^1, ..., 2^{p-2}} can be multiplied by some permutation of 1 to p-1 and summed to 0 modulo p.Alternatively, perhaps using the fact that the numbers 1 to p-1 modulo p form a multiplicative group, which is cyclic. So if we can find a permutation such that the sum is 0, then the answer is yes.Alternatively, maybe we can treat this as a system of equations. But since it's a single equation modulo p, and we have (p-1)! possible permutations, it's likely that for some primes, such a permutation exists.Alternatively, consider the following. Since the weights are 1, 2, 4, ..., 2^{p-2}, and the numbers are 1 to p-1. If the set {1, 2, 4, ..., 2^{p-2}} forms a basis for the vector space GF(p)^{p-1}, then we can find coefficients (the numbers 1 to p-1) such that the sum is zero. However, GF(p) is a field, but the vector space dimension is 1 here, since we are dealing with a single equation.Wait, maybe not. Let me think differently. The equation is:a_1 * 1 + a_2 * 2 + ... + a_{p-1} * 2^{p-2} ≡ 0 mod p,where each a_i is a unique number from 1 to p-1. So we need to assign the numbers 1 to p-1 to the coefficients a_i such that the above holds.Alternatively, think of the sum as the dot product of the vector (a_1, a_2, ..., a_{p-1}) and the vector (1, 2, 4, ..., 2^{p-2}) modulo p. So we need this dot product to be 0.Given that the numbers a_i are a permutation of 1 to p-1, the question is whether the hyperplane defined by the vector (1, 2, 4, ..., 2^{p-2}) contains any permutation of the standard basis vectors.Alternatively, since permutations are invertible, maybe we can rephrase this as: does there exist a permutation matrix P such that (1, 2, 4, ..., 2^{p-2}) * P * (1, 2, ..., p-1)^T ≡ 0 mod p. But I'm not sure if this helps.Alternatively, consider generating all possible sums. For a given prime p, the number of possible sums modulo p is at most p-1, but since we have (p-1)! permutations, by the pigeonhole principle, many permutations will result in the same sum modulo p. However, we need at least one of these sums to be 0.But does 0 necessarily appear among these sums? It's not clear. It might depend on the prime p.Alternatively, perhaps using the fact that the multiplicative group modulo p is cyclic, generated by some primitive root g. Then, the numbers 1, 2, ..., p-1 can be expressed as powers of g. Suppose we write each number as g^k for k from 0 to p-2. Then the sum becomes sum_{k=0}^{p-2} 2^k * g^{sigma(k)}, where sigma is a permutation of 0 to p-2. We need this sum to be 0 modulo p.This might not be helpful directly, but maybe using properties of cyclotomic polynomials or Gauss sums. However, this seems too advanced for an initial problem-solving approach.Alternatively, consider small primes and look for a pattern.We saw that p=2 and p=3 are impossible. Let's try p=5.For p=5, we need to find a permutation of 1,2,3,4 such that 1*2^3 + 2*2^2 + 3*2^1 + 4*2^0 ≡ 0 mod 5. Wait, but hold on, the coefficients are 2^{p-1 - i} for the i-th term in the permutation. Wait, no. The first removed number is multiplied by 2^{3} (since p-1=4, 4-1=3), the second by 2^{2}, etc. But if we permute the numbers, then the weight depends on the position in the removal order.Wait, maybe this approach is correct. Let's formalize it.If we remove the numbers in the order x_1, x_2, x_3, x_4, then the final R is:x_1*2^{3} + x_2*2^{2} + x_3*2^{1} + x_4*2^{0}.Because:- The first removed number x_1 is added to R once and to the other three numbers. Then, when the second number x_2 is removed, it has already been increased by x_1, so x_2 contributes x_1 + x_2 to R. But wait, no. Wait, let's track the exact contributions.Wait, perhaps the coefficient of each x_i is actually 2^{k}, where k is the number of times it's been added to subsequent numbers. But this might not be straightforward.Alternatively, let's take p=5 and try different permutations.Case 1: Remove in order 1,2,3,4.Compute R:First removal: 1. Add 1 to R and to 2,3,4. R=1. L becomes 3,4,5 (i.e., 2+1=3, 3+1=4, 4+1=5).Second removal: 2 (but now it's 3). Add 3 to R and to 4,5. R=1+3=4. L becomes 7,8 (4+3=7, 5+3=8).Third removal: 3 (now 7). Add 7 to R and to 8. R=4+7=11. L becomes 15 (8+7=15).Fourth removal: 4 (now 15). Add 15 to R. R=11+15=26.26 mod 5 is 1. Not 0.Case 2: Remove in order 4,3,2,1.First removal:4. Add 4 to R and to 1,2,3. R=4. L becomes 5,6,7.Second removal:3 (now 7). Add 7 to R and to 5,6. R=4+7=11. L becomes 12,13.Third removal:2 (now 13). Add 13 to R and to 12. R=11+13=24. L becomes 25.Fourth removal:1 (now 25). Add 25 to R. R=24+25=49. 49 mod 5 is 4. Not 0.Hmm, not working. Let's try another permutation. How about removing 2,4,1,3.First removal:2. Add 2 to R and to 1,3,4. R=2. L becomes 3,5,6.Second removal:4 (now 6). Add 6 to R and to 3,5. R=2+6=8. L becomes 9,11.Third removal:1 (now 9). Add 9 to R and to 11. R=8+9=17. L becomes 20.Fourth removal:3 (now 20). Add 20 to R. R=17+20=37. 37 mod 5 is 2. Not 0.Hmm. Another permutation: 1,3,4,2.First removal:1. R=1. L becomes 3,4,5.Second removal:3 (now 4). Add 4 to R and to 5,3. Wait, no, after first removal, L is 3,4,5. Removing 3: add 3 to R and to 4,5. So R=1+3=4. L becomes 7,8.Third removal:4 (now 7). Add 7 to R and to 8. R=4+7=11. L becomes 15.Fourth removal:2 (now 15). Add 15 to R. R=11+15=26. 26 mod 5 is 1. Not 0.This is frustrating. Maybe p=5 is also impossible? Or maybe I need a different approach.Alternatively, perhaps the final R is equal to the sum of the numbers multiplied by 2^{p-1 -1} ?Wait, in p=3, sum is 1+2=3. 3*2^{1} =6. 6 mod 3=0. But earlier calculation showed R=4 or 5, which are not 0. So this seems contradictory. Therefore, this approach is wrong.Wait, what's the relation between the sum of the numbers and the final R? Let's see.For p=3, the sum of numbers is 3. The final R is 4 or 5, which are 1 and 2 mod 3. So not 0.For p=5, sum of numbers is 10. If we compute 10*2^{4-1}=10*8=80. 80 mod 5=0. But in actual calculations, R was 26 or 37, which mod 5 are 1 and 2. So this also doesn't hold.Therefore, this line of thinking is incorrect. Maybe we need to find another way.Wait, let's try to model the process as a product of matrices or something. Each operation can be represented as a linear transformation. Since we are dealing with linear operations (adding a multiple of one number to others), maybe we can represent each move as a matrix and then the composition of these matrices will give the final result.But since each move affects multiple numbers, this might get complex. Alternatively, think of the entire process as a sequence of operations where each number removed contributes to R and modifies the remaining numbers.Alternatively, consider that each time you remove a number, you're effectively doubling the remaining numbers. Wait, let's see.Suppose we have a number x remaining. When we remove another number y, we add y to x, so x becomes x + y. If we consider that this is equivalent to x + y = x(1) + y(1), but there's a multiplicative effect over multiple steps. Wait, maybe not.Alternatively, think recursively. Suppose we have k numbers left. The next number we remove will be added to the remaining (k-1) numbers and to R. So the total contribution to R is the sum of all removed numbers. But the remaining numbers are each increased by the sum of all previously removed numbers. Therefore, when we remove a number x, it's equal to the original x plus the sum of all previously removed numbers. Therefore, the total R is the sum_{i=1}^{p-1} (x_i + sum_{j=1}^{i-1} x_j).Wait, this is a key insight. Let me formalize it.Let’s denote the order of removal as x_1, x_2, ..., x_{p-1}. Then, when we remove x_i, it has been increased by all previous removals x_1, x_2, ..., x_{i-1}. Therefore, the value added to R when removing x_i is equal to x_i + sum_{j=1}^{i-1} x_j.Therefore, the total R is equal to sum_{i=1}^{p-1} [x_i + sum_{j=1}^{i-1} x_j] = sum_{i=1}^{p-1} x_i + sum_{i=1}^{p-1} sum_{j=1}^{i-1} x_j.Let's compute this:First term: sum_{i=1}^{p-1} x_i = S.Second term: sum_{i=2}^{p-1} sum_{j=1}^{i-1} x_j = sum_{j=1}^{p-2} x_j * (p - 1 - j).Because for each x_j, it appears in the inner sum for all i > j. So x_j is added (p - 1 - j) times.Therefore, R = S + sum_{j=1}^{p-2} x_j * (p - 1 - j).But since x_{p-1} is the last term, it doesn't contribute to the second sum. Therefore:R = sum_{i=1}^{p-1} x_i + sum_{j=1}^{p-2} x_j * (p - 1 - j).Alternatively, R = x_{p-1} + sum_{j=1}^{p-1} x_j * (1 + (p - 1 - j)) where the last term x_{p-1} has coefficient 1 since it doesn't contribute to the second sum.Wait, this might not be the easiest way. Let's denote k = p - 1.Then R = sum_{i=1}^k x_i + sum_{j=1}^{k-1} x_j * (k - j).But since k = p -1,R = sum_{i=1}^{p-1} x_i + sum_{j=1}^{p-2} x_j * (p - 1 - j).Therefore, R = x_{p-1} + sum_{j=1}^{p-1} x_j * (1 + (p -1 - j)) for j < p-1.Wait, no. Let me reindex the second sum. Let m = j, so:sum_{m=1}^{p-2} x_m * (p -1 - m).Therefore, R = sum_{m=1}^{p-1} x_m + sum_{m=1}^{p-2} x_m * (p -1 - m).= x_{p-1} + sum_{m=1}^{p-2} x_m * [1 + (p -1 - m)].= x_{p-1} + sum_{m=1}^{p-2} x_m * (p - m).Thus, R = x_{p-1} + sum_{m=1}^{p-2} x_m * (p - m).But since p is prime, and we are working modulo p, we can reduce coefficients modulo p.Note that p - m ≡ -m mod p. Therefore, R ≡ x_{p-1} + sum_{m=1}^{p-2} x_m * (-m) mod p.So R ≡ - sum_{m=1}^{p-2} m * x_m + x_{p-1} mod p.But the x_i's are a permutation of 1, 2, ..., p-1. Let's denote x_{p-1} as y, and x_1, ..., x_{p-2} as the remaining numbers. Then:R ≡ - sum_{m=1}^{p-2} m * x_m + y mod p.But y is the remaining number not in x_1, ..., x_{p-2}. Since the x_i's are a permutation, y = (sum_{n=1}^{p-1} n) - sum_{m=1}^{p-2} x_m.The sum of numbers 1 to p-1 is S = (p-1)p/2. Therefore, y = S - sum_{m=1}^{p-2} x_m.Substituting back into R:R ≡ - sum_{m=1}^{p-2} m * x_m + S - sum_{m=1}^{p-2} x_m mod p.= S - sum_{m=1}^{p-2} (m * x_m + x_m) mod p.= S - sum_{m=1}^{p-2} x_m (m + 1) mod p.= S - sum_{m=1}^{p-2} x_m (m + 1) mod p.But S = (p-1)p/2. Since p is prime and greater than 2 (for odd primes), S = (p-1)p/2 ≡ 0 mod p, because p divides p. Therefore, S ≡ 0 mod p.Therefore, R ≡ - sum_{m=1}^{p-2} x_m (m + 1) mod p.We need R ≡ 0 mod p, so:sum_{m=1}^{p-2} x_m (m + 1) ≡ 0 mod p.But the x_m's are a permutation of the numbers 1 to p-1 excluding y. However, since x_m are the first p-2 numbers removed, and y is the last one, the x_m's are all numbers from 1 to p-1 except y.Wait, but this complicates things because y is dependent on the x_m's. However, since y = S - sum x_m, and S ≡ 0 mod p, then sum x_m ≡ -y mod p.But I'm not sure if this helps. Let's think differently.We need sum_{m=1}^{p-2} x_m (m + 1) ≡ 0 mod p.But the x_m's are a permutation of the numbers 1 to p-1 excluding y. So, in other words, the x_m's are a subset of size p-2 from 1 to p-1. However, this seems challenging.Alternatively, perhaps consider that the sum sum_{m=1}^{p-2} x_m (m + 1) is equivalent to choosing a subset of p-2 numbers from 1 to p-1 and weighting them by m+1. We need this weighted sum to be 0 mod p.But with the freedom to choose any subset of p-2 numbers (since y can be any number), and permuting them (since the order of removal affects the weighting), maybe we can choose y such that the weighted sum is 0.Alternatively, note that m ranges from 1 to p-2. So m + 1 ranges from 2 to p-1. Therefore, the weights are 2, 3, ..., p-1. So we need to assign numbers from 1 to p-1 (excluding y) to positions with weights 2 to p-1, such that the sum is 0 mod p.This seems like a system of equations, but it's underdetermined. However, since we can choose y freely, perhaps there exists a y such that when we exclude y, the remaining numbers can be arranged in some order to make the sum 0.Alternatively, consider the total sum over all possible subsets. For example, the total sum over all possible y would be sum_{y=1}^{p-1} [sum_{x_m ≠ y} x_m (m + 1)]. But this might not be helpful.Alternatively, think of choosing y such that the sum over the other numbers multiplied by their weights plus something equals 0 mod p.But this is getting too vague. Let's try for p=5.For p=5, we need sum_{m=1}^{3} x_m (m + 1) ≡ 0 mod 5. Here, m ranges from 1 to 3 (p-2=3). The weights are m+1=2,3,4.The x_m's are three numbers from 1,2,3,4, excluding y. So for each possible y (1,2,3,4), we need to check if the remaining three numbers can be assigned to the weights 2,3,4 such that 2x_1 + 3x_2 + 4x_3 ≡ 0 mod 5.Let's try y=1:x_m's are 2,3,4. Assign to weights 2,3,4. Possible permutations:- 2*2 + 3*3 + 4*4 = 4 + 9 + 16 = 29 ≡ 4 mod 5.- 2*2 + 3*4 + 4*3 = 4 + 12 + 12 = 28 ≡ 3 mod 5.- 2*3 + 3*2 + 4*4 = 6 + 6 + 16 = 28 ≡ 3 mod 5.- 2*3 + 3*4 + 4*2 = 6 + 12 + 8 = 26 ≡ 1 mod 5.- 2*4 + 3*2 + 4*3 = 8 + 6 + 12 = 26 ≡ 1 mod 5.- 2*4 + 3*3 + 4*2 = 8 + 9 + 8 = 25 ≡ 0 mod 5.Ah! Here, when y=1, and x_m's are 2,3,4 assigned as x_1=4, x_2=3, x_3=2:Sum is 2*4 + 3*3 + 4*2 = 8 + 9 + 8 = 25 ≡ 0 mod 5.So this works! Therefore, for p=5, by choosing y=1 and arranging the remaining numbers 4,3,2 in that order, we get the sum 25, which is 0 mod 5. Therefore, R ≡ 0 mod 5. Thus, p=5 is possible.Wow, so it's possible for p=5. Therefore, there must be primes where it's possible and others where it's not. Now, let's see what's special about p=5.The key was that the weights 2,3,4 (for p=5) multiplied by some permutation of the remaining numbers gave a sum divisible by p. So, in this case, y=1, and the permutation of 4,3,2 with weights 2,3,4 gave 25, which is 0 mod 5.Now, let's check for p=7.This will be time-consuming, but let's attempt it.For p=7, the weights are m+1 for m=1 to 5 (since p-2=5). So weights are 2,3,4,5,6.We need to assign x_m's as five numbers from 1 to 6, excluding y, such that 2x_1 + 3x_2 + 4x_3 + 5x_4 + 6x_5 ≡ 0 mod 7.We need to check if for some y, the remaining numbers can be permuted to satisfy this.This is tedious, but let's choose y=1. Then numbers are 2,3,4,5,6.We need to find a permutation of these numbers such that 2a + 3b + 4c + 5d + 6e ≡ 0 mod 7.Let me try some combinations.First, assign the largest numbers to the largest coefficients to maximize the sum, then see if it's 0 mod 7.6*6 + 5*5 + 4*4 + 3*3 + 2*2 = 36 + 25 + 16 + 9 + 4 = 90. 90 mod 7 is 90 - 12*7=90-84=6. Not 0.Alternatively, try to balance positive and negative mod 7.For example, 6 is -1 mod 7, 5 is -2, 4 is -3, etc. Let's see:Let me try assigning 6 (which is -1) to the largest coefficient 6 (which is -1):6*6 = (-1)*(-1) = 1 mod 7.5*5: assign 5 (which is -2) to 5 (which is -2): (-2)*(-2)=4 mod 7.4*4: assign 4 (which is -3) to 4 (which is -3): (-3)*(-3)=9≡2 mod7.3*3: assign 3 (which is 3) to 3 (which is 3): 3*3=9≡2 mod7.2*2: assign 2 to 2: 2*2=4 mod7.Total sum:1 +4 +2 +2 +4=13≡6 mod7. Not 0.Alternatively, swap some numbers.Let me try:Assign 2 to 6: 2*6=12≡5 mod7.3 to 5: 3*5=15≡1 mod7.4 to4:4*4=16≡2 mod7.5 to3:5*3=15≡1 mod7.6 to2:6*2=12≡5 mod7.Total sum:5+1+2+1+5=14≡0 mod7. Hey, that works!So permutation: x_1=6, x_2=5, x_3=4, x_4=3, x_5=2. Wait, but we excluded y=1, so numbers are 2,3,4,5,6.Wait, the weights are 2,3,4,5,6 for m=1 to 5.But the assignment here is:x_1 (weight 2): 6x_2 (weight 3):5x_3 (weight4):4x_4 (weight5):3x_5 (weight6):2Then, sum is 2*6 +3*5 +4*4 +5*3 +6*2 =12 +15 +16 +15 +12= 70. 70 mod7=0. Perfect.Therefore, for p=7, by choosing y=1 and arranging the remaining numbers in the order 6,5,4,3,2, we get the sum 70, which is 0 mod7. So p=7 is possible.This suggests that for primes p > 2, it's possible. But wait, what about p=3 and p=2?For p=3, we saw it's impossible. p=2 is also impossible. So maybe the answer is all odd primes except p=2. But p=3 is also impossible. So maybe primes congruent to 1 mod something?Wait, no. Wait, p=5 and p=7 work. p=3 doesn't. Let's check p=11.But this would take too long. Perhaps there's a general pattern. From the examples of p=5 and p=7, it seems that if we can set the numbers in decreasing order multiplied by increasing weights, we get a multiple of p. For p=5, 2*4 +3*3 +4*2= 8+9+8=25. For p=7, 2*6 +3*5 +4*4 +5*3 +6*2=12+15+16+15+12=70. Both 25 and 70 are multiples of p=5 and p=7 respectively.Notice that for p=5, the sum is 25=5*5, and for p=7, the sum is 70=7*10. So the sum is p*(sum of weights)/something.Wait, for p=5, weights are 2,3,4. The sum of weights is 2+3+4=9. The total sum is 25=5*5. For p=7, weights are 2,3,4,5,6. Sum of weights=20. Total sum=70=7*10.So for p=5, 25=5*5, which is p*(sum of weights)/ (something). Sum of weights=9. 9/5=1.8, which doesn't help. For p=7, 70=7*10, sum of weights=20. 20/7≈2.857. Not sure.Alternatively, note that when you assign the numbers in decreasing order to the weights, the sum becomes symmetric. For example, in p=5, assigning 4,3,2 to weights 2,3,4: the weights and numbers are symmetric around the center. Similarly for p=7.This symmetry might be key. If we let x_m = p - m - 1, then perhaps the sum becomes something like (p-1)(p)/2, but multiplied by something. However, this is speculative.Alternatively, the sum when assigning the numbers in reverse order is sum_{k=1}^{p-2} (p - k) * (k + 1). Let's compute this sum:sum_{k=1}^{p-2} (p - k)(k + 1) = sum_{k=1}^{p-2} (p(k + 1) - k(k + 1)).= p sum_{k=1}^{p-2} (k + 1) - sum_{k=1}^{p-2} k(k + 1).First term: p sum_{k=2}^{p-1} k = p [sum_{k=1}^{p-1} k - 1] = p [ (p-1)p/2 - 1 ].Second term: sum_{k=1}^{p-2} k(k + 1) = sum_{k=1}^{p-2} (k^2 + k) = sum_{k=1}^{p-2} k^2 + sum_{k=1}^{p-2} k.= [ (p-2)(p-1)(2p-3)/6 ] + [ (p-2)(p-1)/2 ].This seems complicated, but perhaps simplifying:Let’s compute the first term:p [ (p-1)p/2 - 1 ] = p(p-1)p/2 - p = (p^3 - p^2)/2 - p.Second term:= [ (p-2)(p-1)(2p-3)/6 + (p-2)(p-1)/2 ].Factor out (p-2)(p-1)/6:= (p-2)(p-1)/6 [2p - 3 + 3] = (p-2)(p-1)/6 * 2p = (p-2)(p-1)p/3.Therefore, the total sum is:(p^3 - p^2)/2 - p - (p-2)(p-1)p/3.Simplify:Let’s compute each term:Term1: (p^3 - p^2)/2 - p = (p^3 - p^2 - 2p)/2.Term2: -(p-2)(p-1)p/3.Combine both terms:Total sum = (p^3 - p^2 - 2p)/2 - (p-2)(p-1)p/3.To combine, find common denominator 6:= 3(p^3 - p^2 - 2p)/6 - 2(p-2)(p-1)p/6.= [3p^3 - 3p^2 -6p - 2p(p^2 -3p +2)] /6.Expand the second part:-2p(p^2 -3p +2) = -2p^3 +6p^2 -4p.So numerator:3p^3 -3p^2 -6p -2p^3 +6p^2 -4p = (3p^3 -2p^3) + (-3p^2 +6p^2) + (-6p -4p) = p^3 +3p^2 -10p.Thus, total sum = (p^3 +3p^2 -10p)/6.Factor numerator:p(p^2 +3p -10)/6.Factor quadratic:p^2 +3p -10 = (p+5)(p-2). So total sum = p(p+5)(p-2)/6.Thus, the sum when assigning numbers in reverse order is p(p+5)(p-2)/6.We need this sum to be divisible by p. Since p is prime, the sum is divisible by p if (p+5)(p-2)/6 is an integer.Simplify (p+5)(p-2) = p^2 +3p -10.So the sum is p(p^2 +3p -10)/6.Therefore, the sum is divisible by p, so the term (p^2 +3p -10)/6 must be an integer.Thus, for the sum to be divisible by p, we need (p^2 +3p -10) divisible by 6.So p^2 +3p -10 ≡ 0 mod 6.Let’s check this congruence for primes p.For p=2: 4 +6 -10=0. 0≡0 mod6. But p=2 is not applicable here since we considered p≥5 in this calculation. However, for p=5:25 +15 -10=30. 30/6=5. Integer. p=5 works.For p=7: 49 +21 -10=60. 60/6=10. Integer. p=7 works.For p=11: 121 +33 -10=144. 144/6=24. Integer. p=11 works.For p=3: 9 +9 -10=8. 8≡2 mod6. Not divisible by6. So p=3 doesn't work.For p=13: 169 +39 -10=198. 198/6=33. Integer. p=13 works.Therefore, for primes p≥5, the sum is divisible by 6, hence the total sum is divisible by p. Therefore, for primes p≥5, assigning the numbers in reverse order to the weights gives R=0 mod p.But for p=3, the sum is 8, which is not divisible by6, hence R is not divisible by3.This suggests that for primes p≥5, it is possible to choose the numbers in such a way that the final number is divisible by p, but for p=2 and p=3, it's not possible.Therefore, the answer is all primes p≥5.Final AnswerThe primes are those congruent to boxed{p equiv 1 pmod{2}}, but specifically, all primes except 2 and 3. However, based on the analysis, it's all primes greater than 3. But correcting based on the problem statement and standard answer conventions, the answer is all primes except 2 and 3. Wait, but in the problem statement, p is a prime, and we need to determine for which primes it's possible. From our analysis, p=5 and p=7 work, p=3 and p=2 do not. Hence, the answer is all primes p ≥ 5.But in the box, we need to write the specific answer. According to mathematical problem conventions, likely the answer is all primes congruent to 1 mod something, but our analysis shows it's all primes p ≥5. However, since primes are either 2, or odd primes. But since 2 and 3 are exceptions, the answer should be all primes p where p ≠ 2 and p ≠3. However, in the problem statement, p is a prime, so need to specify for which primes it's possible. The correct answer is all primes p ≥5, or equivalently, all primes p ≠2,3. But since primes are 2,3,5,7,11,..., the answer is all primes p ≥5. However, in the format requested, we need to use a mathematical expression. Usually, primes p ≥5 can be expressed as p ≠2,3. But I need to check.Alternatively, since p=5 works, p=7 works, and primes greater than 7 also work, the answer is all primes p ≥5. But how to express this in a boxed answer. The standard way is to state all primes not equal to 2 or 3. But perhaps in the problem's context, the answer is all primes congruent to 1 mod 4 or something else. But based on our analysis, it's all primes p ≥5.Alternatively, based on the pattern we saw that for primes p ≥5, the sum can be arranged to be divisible by p. Therefore, the answer is all primes p ≥5. Thus, the answer is all primes p ≠2,3. But how to write this.But in the problem statement, it's possible that for all primes p ≡1 mod something, but our examples include p=5≡1 mod4 and p=7≡3 mod4, so it's not congruence. Therefore, the answer is all primes except p=2 and p=3. So the final answer is:boxed{p neq 2 text{ and } p neq 3}But usually, in math problems, such answers are written as "all primes p ≥5", but depending on the problem's expectation. However, since p=2 and p=3 are primes and specifically excluded, it's better to state it as p ≠2,3. However, I need to check the standard answer for this type of problem.Wait, let me verify once more:For p=5: possible.For p=7: possible.For p=11: possible.For p=2,3: impossible.Thus, the answer is all primes except 2 and 3. So the boxed answer should be:boxed{p geq 5}But another way is to state:boxed{p neq 2 text{ and } p neq 3}But in many math competitions, they prefer the former. However, given that the problem is in Chinese (LX OM), and given the context, I think specifying p ≥5 is acceptable. But let me check once more.Wait, in our analysis, the crucial point was that for primes p ≥5, the sum when arranging the numbers in reverse order gives a multiple of p. This works because of the symmetry and the mathematical properties. Therefore, the answer is all primes p ≥5.Final Answerboxed{p geq 5}</think>Given a prime number ( p ), we need to determine for which values of ( p ) it is possible to select numbers from the left side of the board (initially ( 1, 2, ldots, p-1 )) in such a way that the remaining number on the board after ( p-1 ) moves is divisible by ( p ).1. Understanding the Process: - Each move involves selecting a number from the left side, adding it to all remaining numbers on the board, and then erasing the selected number. - The goal is to determine if the final remaining number (on the right side) is divisible by ( p ).2. Key Observations: - The final number on the right side is the sum of all selected numbers, each multiplied by a certain weight based on their order of removal. - The weights are powers of 2, leading to a linear combination of the original numbers modulo ( p ).3. Analysis for Small Primes: - For ( p = 2 ) and ( p = 3 ), it was found that the final number is not divisible by ( p ). - For ( p = 5 ) and ( p = 7 ), it was possible to find a permutation of the numbers such that the final number is divisible by ( p ).4. General Case Analysis: - For primes ( p geq 5 ), it was shown that arranging the numbers in reverse order (decreasing order) with corresponding weights results in a sum that is divisible by ( p ). - This is due to the symmetry and properties of the weights and numbers modulo ( p ).5. Conclusion: - The analysis showed that for primes ( p geq 5 ), the final number can be made divisible by ( p ), while for ( p = 2 ) and ( p = 3 ), it is not possible.Thus, the final answer is:[boxed{p geq 5}]]

🔑:Answer: For all prime numbers p except 2 and 3.Let t_i for i=1,2, cdots,p-1 denote the initial value of the number thatwas erased from the board in the i -th move. Thus, the sequence (t_1,t_2,t_3, cdots,t_{p-1}) is a permutationof the sequence (1,2,3, cdots,p-1) and uniquely determines the way the erased numbers are chosen in subsequent moves.Let us introduce the notationWe will prove by induction that after the i -th move, all numbers on the board are greaterby s_i than their initial values.Indeed: this statement is true for i = 1 , since in the first move the number t_1 was erased, increasing all other numbers by s_1 = t_1 . If, however, afterthe i -th move all numbers on the board were greater by s_i than their initial values,then in particular the number initially equal to t_{i+1} had the value t_{i+1} + s_i after the i -th move. Therefore, as a result of the (i+1) -th move, the amount t_{i+1} + s_i was added to all remaining numbers. Since the numbers were greater by s_i than their initial values after the i -th move,they became greater by t_{i+1}+s_i+s_i than their initial values after the (i+1) -th move.It remains to observe thatwhich completes the inductive proof. We have thus shown that after all p moves, the numberremaining on the right side of the board isWe have thus reduced the problem to finding all prime numbers p for which there exists a permutation (t_1,t_2,t_3, cdots,t_{p-1}) of the sequence (1,2,3, cdots,p-1) such that the number (1) is divisible by p .We directly check that for p=2 and p=3 it is not possible to obtain a number divisible by p :for p = 2 the number (1) is 1 , and for p = 3 it can be 1+2cdot 2=5 or 2+2cdot 1 = 4 .We will show that if p is a prime number greater than 3 , then for the permutationformed from the sequence (1,2,3,4, cdots,p - 2,p - 1) by swapping the positions in pairs (2j-1,2j) for j geqslant 2 and then reversing the order, the number (1) is divisible by p .Indeed, for the permutation (2), the number (1) has the valueFor j =2,3, cdots, frac{1}{2}(p-1) , the relationholds, from which it follows that L = M - N , where M is the sum of numbers of the form 2^{i-1}i for i =1,2, cdots,p-1 , and N =2^2 +2^4 +2^6 + cdots +2^{p-3} . Using the formula for the sum of consecutiveterms of a geometric sequence, we calculate thatandThus,By Fermat's Little Theorem (see LI Mathematical Olympiad, Main Committee Report,Warsaw 2001, Appendix A, p. 102), the number 2^{p-1} -1 is divisible by p . Since p is a prime number greater than 3 , the fraction appearing on the right side of relation (3) is divisible by p . Therefore, the right side of this relation is divisible by p , which completesthe solution of the problem.

🧠:[<think>Okay, let me see. The problem says that if x=2 is one of the roots of the equation x² - b x + c = 0, then we need to find the value of 4b - 2c + 1. Hmm, I need to figure out how to use the fact that 2 is a root to find the relationship between b and c, and then plug that into the expression 4b - 2c + 1. Let me start by recalling some algebra.First, in a quadratic equation of the form x² + px + q = 0, the sum of the roots is -p and the product is q. Wait, but here the equation is written as x² - b x + c = 0. So comparing that to the standard form x² + px + q = 0, it looks like p here is -b and q is c. So the sum of the roots should be -p, which would be -(-b) = b, and the product is q, which is c. So for the equation x² - b x + c = 0, the sum of the roots is b and the product is c.Since 2 is one of the roots, let me call the other root r. Then according to Vieta's formulas, the sum of the roots is 2 + r = b, and the product is 2 * r = c. So if I can express r in terms of b or c, I can relate b and c. But since we have two equations here, maybe I can solve for one variable in terms of the other.So let me write down the two equations:1) 2 + r = b2) 2 * r = cFrom equation 1, I can solve for r: r = b - 2.Then substitute this into equation 2: 2*(b - 2) = c. So 2b - 4 = c. Therefore, c = 2b - 4.So now, c is expressed in terms of b. The problem asks for 4b - 2c + 1. Let's substitute c with 2b - 4.So 4b - 2*(2b - 4) + 1. Let's compute this step by step.First, compute 2*(2b - 4): that's 4b - 8.Then subtract that from 4b: 4b - (4b - 8). Which is 4b - 4b + 8 = 8.Then add 1: 8 + 1 = 9.Wait, that seems straightforward. So 4b - 2c + 1 equals 9. Let me check my steps again to make sure I didn't make a mistake.First, since x=2 is a root, plugging x=2 into the equation should satisfy it. Let me verify that. If x=2, then the equation becomes 2² - b*2 + c = 0. Which is 4 - 2b + c = 0. So 4 - 2b + c = 0. Therefore, c = 2b - 4. Which matches what I had before. So that's correct.Alternatively, using Vieta's formulas, as I did earlier, gives the same result. So that's good.Then substituting c into the expression 4b - 2c +1:4b - 2*(2b - 4) +1 = 4b - 4b + 8 +1 = 9. Yes, that seems right.Alternatively, maybe there's another way to think about it. Let's see. If 2 is a root, then the quadratic can be factored as (x - 2)(x - r) = 0, where r is the other root. Expanding that gives x² - (2 + r)x + 2r = 0. Comparing that to the original equation x² - b x + c = 0, we can see that b = 2 + r and c = 2r. So again, same equations as before.Therefore, expressing r as b - 2 and substituting into c = 2r gives c = 2(b - 2) = 2b - 4. Therefore, substituting into 4b - 2c +1:4b - 2*(2b - 4) +1 = 4b -4b +8 +1 = 9.Yes, same result. So seems like 9 is the answer.But let me just try plugging in numbers. Suppose we pick a value for b, find c, compute 4b -2c +1, and see if it's always 9. Let's test with an example.Let me choose b = 3. Then from c = 2b -4, c = 6 -4 = 2. So the equation is x² -3x +2 =0. The roots are x=1 and x=2, which works. Then 4b -2c +1 = 4*3 -2*2 +1 =12 -4 +1=9. Correct.Another example. Let me choose b=5. Then c=2*5 -4=10-4=6. So equation is x² -5x +6=0, roots x=2 and x=3. Then 4*5 -2*6 +1=20 -12 +1=9. Also works.Another example, b=4. Then c=2*4 -4=4. Equation x² -4x +4=0, which factors as (x-2)^2=0, so double root at x=2. Then 4*4 -2*4 +1=16 -8 +1=9. Also correct.So regardless of the value of b (as long as 2 is a root), 4b -2c +1 is always 9. Therefore, the answer is 9.Final Answerboxed{9}</think>Given that ( x = 2 ) is one of the roots of the quadratic equation ( x^2 - bx + c = 0 ), we need to find the value of ( 4b - 2c + 1 ).First, we use Vieta's formulas for the roots of the quadratic equation. If the roots are ( 2 ) and ( r ), then:- The sum of the roots is ( 2 + r = b )- The product of the roots is ( 2r = c )From the sum of the roots, we solve for ( r ):[ r = b - 2 ]Substituting ( r ) into the product of the roots gives:[ 2(b - 2) = c ][ c = 2b - 4 ]Next, we substitute ( c = 2b - 4 ) into the expression ( 4b - 2c + 1 ):[4b - 2(2b - 4) + 1 = 4b - 4b + 8 + 1 = 9]Thus, the value of ( 4b - 2c + 1 ) is (boxed{9}).,<think>Alright, so I need to figure out the value of 4b - 2c + 1, given that x=2 is a root of the quadratic equation x² - bx + c = 0. Let me start by recalling some basics about quadratic equations and their roots.First, I remember that for any quadratic equation of the form ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a. In this problem, the equation is x² - bx + c = 0. Comparing this to the standard form, the coefficients here are: a = 1 (the coefficient of x²), b = -b (wait, hold on, that might be confusing because the coefficient is actually written as -b in the equation. Let me check that again.)The given equation is x² - bx + c = 0. So, in standard form ax² + bx + c = 0, here a = 1, the coefficient of x is -b, and the constant term is c. Therefore, the sum of the roots would be -(-b)/1 = b, and the product of the roots is c/1 = c. So, sum of roots is b, product is c.We are told that x=2 is one of the roots. Let's call the other root r. Then, according to the sum and product of roots:Sum of roots: 2 + r = bProduct of roots: 2 * r = cSo, the problem is essentially asking for 4b - 2c + 1. But to compute that, we need to find values of b and c. However, we only know one root (2), and the other root is unknown. So, we have two variables here: r, b, and c. But actually, since b and c can be expressed in terms of r, maybe we can express 4b - 2c + 1 in terms of r and see if there's a way to find it without knowing r?Wait, but the problem doesn't give us any other information. It just says x=2 is one of the roots. So, unless there's another condition, perhaps the expression 4b - 2c + 1 can be determined regardless of the value of r? Let me check that.Given that b = 2 + r and c = 2r. Let's substitute these into 4b - 2c + 1:4b - 2c + 1 = 4*(2 + r) - 2*(2r) + 1Let me compute that step by step.First, compute 4*(2 + r): 4*2 + 4*r = 8 + 4rThen compute -2*(2r): -4rSo, adding those together: 8 + 4r - 4r + 1Simplify: 8 + 0 + 1 = 9Wait, that's interesting. The terms with r cancel out. So, regardless of the value of r (the other root), the expression 4b - 2c + 1 equals 9. Therefore, the answer is 9.But let me verify this again to make sure I didn't make a mistake. Let's substitute b and c in terms of r:4b - 2c + 1 = 4*(2 + r) - 2*(2r) + 1Compute each term:4*(2 + r) = 8 + 4r-2*(2r) = -4rThen, +1Adding them all together: 8 + 4r - 4r + 1 = (8 + 1) + (4r - 4r) = 9 + 0 = 9Yes, that seems correct. The terms with r cancel each other out, so the expression simplifies to 9. Therefore, even though we don't know the other root or the specific values of b and c, their combination in this expression results in 9.Alternatively, another way to approach this problem is by plugging x=2 into the equation since it's a root. Let me try that method to confirm.Given that x=2 is a root of x² - bx + c = 0, substituting x=2 into the equation should satisfy it:(2)² - b*(2) + c = 0Which simplifies to:4 - 2b + c = 0So, rearranged: c = 2b - 4Now, we can use this equation (c = 2b - 4) to substitute into the expression 4b - 2c + 1.Substitute c with 2b -4:4b - 2*(2b -4) + 1Compute each term:4b - 4b + 8 + 1Again, 4b -4b cancels out, leaving 8 +1 = 9Same result! So whether I use the relationships from the sum and product of roots or substitute the known root into the equation, I end up with 9.Therefore, the answer is 9.Final AnswerThe value of (4b - 2c + 1) is boxed{9}.</think>Given that (x = 2) is one of the roots of the quadratic equation (x^2 - bx + c = 0), we need to find the value of (4b - 2c + 1).First, we use the fact that substituting (x = 2) into the equation must satisfy it:[(2)^2 - b(2) + c = 0]Simplifying this, we get:[4 - 2b + c = 0 implies c = 2b - 4]Next, we substitute (c = 2b - 4) into the expression (4b - 2c + 1):[4b - 2(2b - 4) + 1]Expanding and simplifying:[4b - 4b + 8 + 1 = 0 + 9 = 9]Thus, the value of (4b - 2c + 1) is (boxed{9}).]

🔑:To solve the problem, we follow these steps:1. Substitute x=2 into the equation x^2 - bx + c = 0:[2^2 - 2b + c = 0][4 - 2b + c = 0]2. Rearrange the equation to solve for 2b - c:[2b - c = 4]3. To find the value of 4b - 2c + 1, we manipulate the expression we found:[4b - 2c + 1 = 2(2b - c) + 1][= 2 times 4 + 1][= 8 + 1][= 9]Therefore, the value of 4b - 2c + 1 is boxed{9}.

🧠:[<think>Okay, so I need to find the range of the function y equals the square root of (x minus 4) plus the square root of (15 minus 3x). Hmm, let's start by understanding the domain of this function first because square roots can't have negative numbers inside them. First, for the square root of (x - 4) to be real, the expression inside the square root has to be non-negative. So x - 4 has to be greater than or equal to 0. That means x ≥ 4. Next, for the square root of (15 - 3x) to be real, 15 - 3x must also be non-negative. Let's solve that: 15 - 3x ≥ 0. If I subtract 15 from both sides, I get -3x ≥ -15. Then, dividing both sides by -3 (and remembering to reverse the inequality sign when dividing by a negative number) gives x ≤ 5.So the domain of the function is all x such that x is between 4 and 5, inclusive. That is, 4 ≤ x ≤ 5. Okay, so x can be 4, 5, or anything in between. Now, the next step is to figure out the range, which means finding all possible values of y given that x is in [4,5].The function is y = sqrt(x - 4) + sqrt(15 - 3x). Let me see. Since both square roots are non-negative, y will be at least 0. But the actual range will depend on how these two square roots behave as x varies between 4 and 5.Maybe I can consider substituting a variable to make this easier. Let me let t = x - 4. Since x is between 4 and 5, t will be between 0 and 1. Then, substituting t into the equation:x = t + 4, so 15 - 3x becomes 15 - 3(t + 4) = 15 - 3t - 12 = 3 - 3t = 3(1 - t). So the second square root becomes sqrt(3(1 - t)).So the function becomes y = sqrt(t) + sqrt(3(1 - t)). So y = sqrt(t) + sqrt(3) * sqrt(1 - t). That might be a bit easier to handle. Maybe I can write this as y = sqrt(t) + sqrt(3) * sqrt(1 - t) where t is between 0 and 1.Alternatively, maybe a trigonometric substitution would work here. The problem mentioned that a trigonometric substitution can be used because x is between 4 and 5, which corresponds to t between 0 and 1. So maybe let t = sin²θ or something like that. Let's see.If I let t = sin²θ, then sqrt(t) = sinθ, and sqrt(1 - t) = cosθ. Then y = sinθ + sqrt(3)cosθ. That seems manageable. Then, since t ranges from 0 to 1, θ would range from 0 to π/2. So y becomes sinθ + sqrt(3)cosθ. Hmm, that's a linear combination of sine and cosine, which can be expressed as a single sine (or cosine) function with a phase shift. The maximum and minimum of such an expression can be found using the amplitude formula.The amplitude of y = A sinθ + B cosθ is sqrt(A² + B²). So here, A is 1 and B is sqrt(3). So the amplitude would be sqrt(1 + 3) = sqrt(4) = 2. Therefore, the maximum value of y would be 2, and the minimum would be -2. But since θ is between 0 and π/2, sinθ and cosθ are both non-negative, so y would be between 0 and 2. But wait, when θ is 0, y = sqrt(3)*cos0 = sqrt(3)*1 = sqrt(3), and sin0 = 0. So y = sqrt(3) ≈ 1.732. When θ is π/3, for example, sinθ = sqrt(3)/2, cosθ = 1/2, so y would be sqrt(3)/2 + sqrt(3)*(1/2) = sqrt(3)/2 + sqrt(3)/2 = sqrt(3). Wait, that's the same as before. Hmm, maybe I need to check when θ varies from 0 to π/2.Wait, if t = sin²θ, then θ ranges from 0 to π/2 because t ranges from 0 to 1. So when θ = 0, t = 0, so x = 4, and y = sqrt(0) + sqrt(3*(1 - 0)) = 0 + sqrt(3) = sqrt(3). When θ = π/2, t = sin²(π/2) = 1, so x = 5, and y = sqrt(1) + sqrt(3*(1 - 1)) = 1 + 0 = 1. So the maximum and minimum might not be given by the amplitude here because θ is restricted. Wait, but earlier I thought that the expression y = sinθ + sqrt(3)cosθ has amplitude 2, but when θ is between 0 and π/2, maybe the maximum is achieved somewhere in between. Let me check.Let me compute the derivative of y with respect to θ. If y = sinθ + sqrt(3)cosθ, then dy/dθ = cosθ - sqrt(3)sinθ. Setting derivative to zero: cosθ - sqrt(3)sinθ = 0 => cosθ = sqrt(3)sinθ => tanθ = 1/sqrt(3) => θ = π/6. So the critical point is at θ = π/6. Let's evaluate y at θ = π/6. sin(π/6) = 1/2, cos(π/6) = sqrt(3)/2. So y = 1/2 + sqrt(3)*(sqrt(3)/2) = 1/2 + (3)/2 = 2. So the maximum value is 2 at θ = π/6, which is t = sin²(π/6) = (1/2)^2 = 1/4. So x = 4 + t = 4.25. Then, the minimum values at the endpoints: θ = 0 gives y = sqrt(3) ≈ 1.732, and θ = π/2 gives y = 1. So the range of y is from 1 to 2. Wait, but sqrt(3) is approximately 1.732, which is larger than 1. So wait, when x is 4, y is sqrt(3), when x is 5, y is 1, and when x is 4.25, y is 2. Therefore, the maximum value of y is 2, and the minimum is 1. So the range is [1, 2].Wait, but let me confirm this with another approach. Maybe squaring both sides. Let me try that. Let's set y = sqrt(x - 4) + sqrt(15 - 3x). Let's square both sides: y² = (sqrt(x - 4) + sqrt(15 - 3x))² = (x - 4) + 2*sqrt{(x - 4)(15 - 3x)} + (15 - 3x). Simplify the terms: x - 4 + 15 - 3x = (x - 3x) + (-4 + 15) = (-2x) + 11. So y² = -2x + 11 + 2*sqrt{(x - 4)(15 - 3x)}. Let's denote S = sqrt{(x - 4)(15 - 3x)}. So y² = -2x + 11 + 2S. Then, let's solve for S: 2S = y² + 2x - 11 => S = (y² + 2x - 11)/2. But S is sqrt{(x - 4)(15 - 3x)}. Let's compute that expression inside the square root.(x - 4)(15 - 3x) = 15x - 3x² - 60 + 12x = (15x + 12x) - 3x² - 60 = 27x - 3x² - 60 = -3x² + 27x - 60. Let's factor that: -3(x² - 9x + 20) = -3(x - 4)(x - 5). Wait, but x is between 4 and 5, so (x - 4) is non-negative, (x - 5) is non-positive, so (x - 4)(x - 5) is non-positive. Therefore, (x - 4)(15 - 3x) = -3(x - 4)(x - 5) = -3*(non-negative)*(non-positive) = non-negative. So inside the square root is non-negative, which makes sense.So S = sqrt{-3(x - 4)(x - 5)}. But we can write that as sqrt{3(4 - x)(x - 5)}? Wait, no. Wait, (x - 4)(15 - 3x) = (x - 4)*3*(5 - x) = 3(x - 4)(5 - x). So S = sqrt{3(x - 4)(5 - x)}. Since x is between 4 and 5, both (x - 4) and (5 - x) are non-negative, so S is real and non-negative. So S = sqrt{3(x - 4)(5 - x)}.So going back to S = (y² + 2x - 11)/2. Therefore, sqrt{3(x - 4)(5 - x)} = (y² + 2x - 11)/2. Let's square both sides again to eliminate the square root:3(x - 4)(5 - x) = [(y² + 2x - 11)/2]^2.Multiply both sides by 4:12(x - 4)(5 - x) = (y² + 2x - 11)².Let me expand the left side:12(x - 4)(5 - x) = 12*(-x² + 9x - 20). Wait, (x - 4)(5 - x) = -x² + 9x - 20. So 12*(-x² + 9x -20) = -12x² + 108x - 240.Now, the right side is (y² + 2x -11)^2. Let's expand that:(y² + 2x -11)^2 = y^4 + 4x² + 121 + 4x y² - 22 y² - 44x.Wait, maybe step by step:Let me let A = y² + 2x -11. Then A² = (y²)^2 + (2x)^2 + (-11)^2 + 2*y²*2x + 2*y²*(-11) + 2*2x*(-11).So that's y^4 + 4x² + 121 + 4x y² - 22 y² - 44x.Therefore, the equation becomes:-12x² + 108x -240 = y^4 + 4x² + 121 + 4x y² - 22 y² -44x.Bring all terms to one side:-12x² +108x -240 - y^4 -4x² -121 -4x y² +22 y² +44x = 0.Combine like terms:(-12x² -4x²) + (108x +44x) + (-240 -121) + (-y^4) + (-4x y²) +22 y² = 0.So:-16x² +152x -361 - y^4 -4x y² +22 y² =0.Hmm, this seems complicated. Maybe this approach is getting too messy. Perhaps going back to the trigonometric substitution was better.Earlier, by substituting t = x -4, then t is in [0,1], and y = sqrt(t) + sqrt(3(1 - t)). Then, using t = sin²θ, which leads to y = sinθ + sqrt(3)cosθ, which we found can reach a maximum of 2 and minimum of 1. Wait, but let me check that again.Wait, when θ is 0, y = sqrt(3) ≈1.732, when θ is π/6, y=2, and when θ is π/2, y=1. So, as θ increases from 0 to π/2, y decreases from sqrt(3) to 1, but reaches a maximum of 2 at θ=π/6. Wait, that seems contradictory. Wait, if θ starts at 0, y is sqrt(3). Then, as θ increases to π/6, y increases to 2. Then, as θ continues to increase to π/2, y decreases to 1. So the maximum value is 2, and the minimum is 1. Therefore, the range of y is [1, 2].Wait, but when x=4, which is θ=0, y=sqrt(3)≈1.732, which is greater than 1. So the minimum value occurs at x=5, y=1. So how come the range is from 1 to 2? Because the function starts at sqrt(3) when x=4, goes up to 2 when x=4.25, then goes back down to 1 when x=5. So the maximum is 2, the minimum is 1. Therefore, the range is [1,2].Let me verify this by testing some values. When x=4, y=sqrt(0) + sqrt(15 - 12)=sqrt(3)≈1.732. When x=5, y=sqrt(1)+sqrt(0)=1. When x=4.25 (which is 17/4), let's compute y. x=4.25, so x-4=0.25, sqrt(0.25)=0.5. 15 -3x=15 -12.75=2.25, sqrt(2.25)=1.5. So y=0.5+1.5=2. That's the maximum. So indeed, the maximum is 2, and the minimum is 1.Alternatively, if I use calculus on the original function. Let me define f(x) = sqrt(x -4) + sqrt(15 -3x). Find its derivative and critical points.f'(x) = (1/(2*sqrt(x -4))) + (-3/(2*sqrt(15 -3x))).Set derivative equal to zero:1/(2*sqrt(x -4)) - 3/(2*sqrt(15 -3x)) =0Multiply both sides by 2:1/sqrt(x -4) - 3/sqrt(15 -3x) =0Bring the second term to the other side:1/sqrt(x -4) = 3/sqrt(15 -3x)Cross multiply:sqrt(15 -3x) = 3 sqrt(x -4)Square both sides:15 -3x = 9(x -4)15 -3x =9x -3615 +36 =9x +3x51=12xx=51/12=17/4=4.25So critical point at x=4.25, which is the same as before. Then, f(4.25)=0.5 +1.5=2, which is the maximum. Then, the endpoints x=4 gives y=sqrt(0)+sqrt(3)=sqrt(3)≈1.732, x=5 gives y=1+0=1. So the minimum is 1, maximum is 2. Therefore, the range is [1,2].Therefore, the answer should be [1,2], written in box notation.Final AnswerThe range of the function is boxed{[1, 2]}.</think>To find the range of the function ( y = sqrt{x - 4} + sqrt{15 - 3x} ), we first determine the domain. The expressions under the square roots must be non-negative, leading to ( 4 leq x leq 5 ).Next, we use a trigonometric substitution to simplify the analysis. Let ( t = x - 4 ), so ( t ) ranges from 0 to 1. The function becomes ( y = sqrt{t} + sqrt{3(1 - t)} ). Substituting ( t = sin^2 theta ), we get ( y = sin theta + sqrt{3} cos theta ).This expression can be rewritten as ( y = 2 sin left( theta + frac{pi}{3} right) ), which has a maximum value of 2 and a minimum value of -2. However, since ( theta ) ranges from 0 to ( frac{pi}{2} ), the valid range of ( y ) is between 1 and 2. To confirm, we check the endpoints and critical points:- When ( x = 4 ), ( y = sqrt{0} + sqrt{15 - 12} = sqrt{3} approx 1.732 ).- When ( x = 5 ), ( y = sqrt{1} + sqrt{0} = 1 ).- The critical point at ( x = 4.25 ) (found by setting the derivative to zero) gives ( y = 0.5 + 1.5 = 2 ).Thus, the range of the function is (boxed{[1, 2]}).,<think>Alright, let's see. I need to find the range of the function y equals the square root of (x minus 4) plus the square root of (15 minus 3x). Hmm, okay. So first, the function is a sum of two square roots. That usually means there are some domain restrictions because the expressions inside the square roots have to be non-negative. Let me start by figuring out the domain because the range will depend on that.So, the first square root is sqrt(x - 4). For this to be real, x - 4 must be greater than or equal to zero. That means x ≥ 4. The second square root is sqrt(15 - 3x). For this one, 15 - 3x has to be ≥ 0. Let me solve that: 15 - 3x ≥ 0 → -3x ≥ -15 → x ≤ 5. So combining both conditions, x has to be between 4 and 5, inclusive. So the domain is [4, 5]. That's the first step.Now, I need to find the range of y over this interval. Since y is the sum of two square roots, which are continuous functions on this interval, the range will be from the minimum value of y to the maximum value of y in [4,5]. So I can find the maximum and minimum values of y in this interval.One approach is to compute y at the endpoints, x = 4 and x = 5, and then check if there's a critical point in between where the function might attain a maximum or minimum. To check for critical points, I can take the derivative of y with respect to x and set it to zero.Let me first compute y at the endpoints. When x = 4: sqrt(4 - 4) + sqrt(15 - 12) = sqrt(0) + sqrt(3) = 0 + √3 ≈ 1.732. When x = 5: sqrt(5 - 4) + sqrt(15 - 15) = sqrt(1) + sqrt(0) = 1 + 0 = 1. So at x = 4, y is √3, and at x = 5, y is 1. So the function decreases from x=4 to x=5? Maybe, but it's possible there's a maximum somewhere in between. Let's check the derivative.Let me find dy/dx. The function is y = sqrt(x - 4) + sqrt(15 - 3x). Let's differentiate term by term.The derivative of sqrt(x - 4) is (1/(2*sqrt(x - 4))). The derivative of sqrt(15 - 3x) is (1/(2*sqrt(15 - 3x)))* (-3) because of the chain rule. So combining:dy/dx = 1/(2*sqrt(x - 4)) - 3/(2*sqrt(15 - 3x)).To find critical points, set dy/dx = 0:1/(2*sqrt(x - 4)) - 3/(2*sqrt(15 - 3x)) = 0.Multiply both sides by 2 to eliminate denominators:1/sqrt(x - 4) - 3/sqrt(15 - 3x) = 0.Move the second term to the other side:1/sqrt(x - 4) = 3/sqrt(15 - 3x).Cross-multiplying:sqrt(15 - 3x) = 3*sqrt(x - 4).Now, square both sides to eliminate the square roots:(15 - 3x) = 9*(x - 4).Simplify the right side: 9x - 36.So left side: 15 - 3x = 9x - 36.Bring all terms to one side:15 - 3x -9x + 36 = 0 → 15 + 36 -12x = 0 → 51 -12x = 0 → 12x = 51 → x = 51/12 = 17/4 = 4.25.So x = 17/4 is a critical point. Let's check if this is within the domain [4,5]. 17/4 is 4.25, which is between 4 and 5. So yes, it's valid.Now, we need to evaluate y at x = 4, 4.25, and 5 to determine which gives the maximum and minimum.We already have y at 4 and 5: √3 ≈ 1.732 and 1.Let's compute y at 17/4 = 4.25.First, compute x -4 = 4.25 -4 = 0.25. sqrt(0.25) = 0.5.Then, 15 -3x = 15 -3*(4.25) = 15 -12.75 = 2.25. sqrt(2.25) = 1.5.So y = 0.5 + 1.5 = 2.0. So y at x=4.25 is 2.0.So comparing the three: at x=4, y≈1.732; at x=4.25, y=2; at x=5, y=1. Therefore, the maximum value is 2, and the minimum is 1. So the range of the function is [1, 2].Wait, but let me verify if there are any other critical points. The derivative was set to zero only once, giving x=4.25. Also, since the function is continuous on [4,5], and differentiable in (4,5), with only one critical point, which gives a maximum. Then, since the function decreases from x=4.25 to x=5 and increases from x=4 to x=4.25? Wait, wait, let me check the derivative's sign.Let me pick a value between 4 and 4.25, say x=4.1. Compute dy/dx at x=4.1:1/(2*sqrt(0.1)) - 3/(2*sqrt(15 - 12.3)) = 1/(2*0.316) - 3/(2*sqrt(2.7)).Calculating:1/(0.632) ≈ 1.581sqrt(2.7) ≈ 1.643, so 3/(2*1.643) ≈ 3/(3.286) ≈ 0.913.Therefore, dy/dx ≈ 1.581 - 0.913 ≈ 0.668. Positive. So derivative is positive before x=4.25, meaning function is increasing.Now pick x=4.5 (after critical point). Compute dy/dx:1/(2*sqrt(0.5)) - 3/(2*sqrt(15 -13.5)) = 1/(2*0.707) - 3/(2*sqrt(1.5)).Compute:1/(1.414) ≈ 0.707sqrt(1.5) ≈1.225, so 3/(2*1.225) ≈ 3/2.45 ≈ 1.224So dy/dx ≈ 0.707 -1.224 ≈ -0.517. Negative. So derivative is negative after x=4.25, meaning function is decreasing.Therefore, the function increases from x=4 to x=4.25, reaching a maximum of 2, then decreases from x=4.25 to x=5, reaching a minimum of 1. Therefore, the range is [1, 2].Alternatively, the problem suggested using a trigonometric substitution. Let me see how that would work.Given that the domain is [4,5], so x is between 4 and 5. Let me let x = 4 + sin²θ, but maybe another substitution. Alternatively, since the interval is 1 unit from 4 to 5, perhaps set x = 4 + t, where t ranges from 0 to 1. Then, substitute t = sin²θ or something similar.But maybe let me see. Let x = 4 + t, so t ∈ [0,1]. Then the function becomes sqrt(t) + sqrt(15 - 3*(4 + t)) = sqrt(t) + sqrt(15 -12 -3t) = sqrt(t) + sqrt(3 - 3t) = sqrt(t) + sqrt(3(1 - t)).So y = sqrt(t) + sqrt(3)sqrt(1 - t), with t ∈ [0,1]. Maybe set t = sin²θ, then sqrt(t) = sinθ, and sqrt(1 - t) = cosθ. Then y = sinθ + sqrt(3)cosθ. Then the expression is a linear combination of sin and cos, which can be written as a single sine function with amplitude sqrt(1 + 3) = 2. So y = 2 sin(θ + 60°), since the coefficients are 1 and sqrt(3), which correspond to angles 30° and 60°. Wait, the amplitude is sqrt(1^2 + (sqrt(3))^2) = sqrt(1 + 3) = 2. And the phase shift would be arctan(sqrt(3)/1) = 60°, so y = 2 sin(θ + 60°). Therefore, the maximum value is 2 and the minimum is -2. But since θ is such that t = sin²θ ∈ [0,1], θ ∈ [0, π/2], because t goes from 0 to1. Then, θ ranges from 0 to π/2, so θ + 60° ranges from 60° to 150°, and sin(θ + 60°) ranges from sin(60°) = sqrt(3)/2 to sin(90°) = 1 and back to sin(150°) = 1/2. Wait, actually, as θ increases from 0 to π/2 (0° to 90°), θ + 60° increases from 60° to 150°, so sin(θ + 60°) increases to 90° (where it is 1) then decreases to 150° (where it is 1/2). Therefore, the maximum value of sin(θ + 60°) is 1, and the minimum is 1/2. Therefore, y = 2 sin(θ + 60°) ranges from 2*(1/2) = 1 to 2*1 = 2. Therefore, the range is [1,2], same as before. So that trigonometric substitution method also gives the same result. That's neat.So both methods confirm that the range is from 1 to 2. Therefore, the answer is [1,2].Final AnswerThe range of the function is boxed{[1, 2]}.</think>To find the range of the function ( y = sqrt{x - 4} + sqrt{15 - 3x} ):1. Determine the domain: - The expressions inside the square roots must be non-negative: - ( x - 4 geq 0 ) implies ( x geq 4 ) - ( 15 - 3x geq 0 ) implies ( x leq 5 ) - Therefore, the domain is ( [4, 5] ).2. Evaluate the function at the endpoints: - At ( x = 4 ): ( y = sqrt{4 - 4} + sqrt{15 - 3 cdot 4} = 0 + sqrt{3} = sqrt{3} approx 1.732 ) - At ( x = 5 ): ( y = sqrt{5 - 4} + sqrt{15 - 3 cdot 5} = sqrt{1} + 0 = 1 )3. Find critical points by differentiating: - The derivative ( frac{dy}{dx} = frac{1}{2sqrt{x - 4}} - frac{3}{2sqrt{15 - 3x}} ) - Set the derivative to zero and solve: [ frac{1}{2sqrt{x - 4}} = frac{3}{2sqrt{15 - 3x}} implies sqrt{15 - 3x} = 3sqrt{x - 4} ] Squaring both sides: [ 15 - 3x = 9(x - 4) implies 15 - 3x = 9x - 36 implies 51 = 12x implies x = frac{17}{4} = 4.25 ] - Evaluate ( y ) at ( x = 4.25 ): [ y = sqrt{4.25 - 4} + sqrt{15 - 3 cdot 4.25} = sqrt{0.25} + sqrt{2.25} = 0.5 + 1.5 = 2 ]4. Verify using trigonometric substitution: - Substitute ( x = 4 + t ) where ( t in [0, 1] ): [ y = sqrt{t} + sqrt{3(1 - t)} ] - Let ( t = sin^2 theta ), then ( y = sin theta + sqrt{3} cos theta ), which simplifies to ( 2 sin(theta + 60^circ) ). The maximum value is 2 and the minimum value is 1.Thus, the range of the function is (boxed{[1, 2]}).]

🔑:Solution: From left{begin{array}{l}x-4 geqslant 0, 15-3 x geqslant 0,end{array}right. we get 4 leqslant x leqslant 5.Let x=4+sin ^{2} thetaleft(0 leqslant theta leqslant frac{pi}{2}right), theny=sin theta+sqrt{3} cos theta=2 sin left(theta+frac{pi}{3}right).Since theta inleft[0, frac{pi}{2}right], therefore,frac{pi}{3} leqslant theta+frac{pi}{3} leqslant frac{5 pi}{6}.Thus, when theta=frac{pi}{2}, y_{text {min }}=1;when theta=frac{pi}{6}, y_{text {max }}=2.Hence, the range of the function is [1,2].Comment: The function after the trigonometric substitution is concise and clear, but it must be particularly noted that the transformed function y=2 sin left(theta+frac{pi}{3}right) is not a monotonic function on left[0, frac{pi}{2}right], so the maximum value cannot be obtained at its left endpoint.

🧠:[<think>Okay, let's see. I need to find the distance from the center of circle C to the line l. The circle C has a polar equation ρ = 4 cos θ, and the line l is given parametrically in Cartesian coordinates. Hmm, okay. Let me start by recalling some concepts.First, the parametric equations of the line l are x = 2t - 1 and y = 4 - 2t, where t is a real number. I might need to convert this parametric form into a standard Cartesian equation for a line, maybe in the form Ax + By + C = 0, which would make it easier to compute the distance from a point to the line later on.To convert the parametric equations to Cartesian form, I can solve for t in one equation and substitute into the other. Let's take the x equation: x = 2t - 1. Solving for t gives t = (x + 1)/2. Then substitute this into the y equation: y = 4 - 2t = 4 - 2*(x + 1)/2. Simplifying that: the 2 in the numerator and denominator cancel out, so y = 4 - (x + 1) = 4 - x - 1 = 3 - x. So, the Cartesian equation of line l is y = -x + 3. Alternatively, rearranged as x + y - 3 = 0. So, A = 1, B = 1, C = -3. Got that.Next, the polar equation of the circle C is ρ = 4 cos θ. I need to find the center of this circle in Cartesian coordinates to then compute the distance from that center to the line l. I remember that in polar coordinates, equations of the form ρ = 2a cos θ represent circles with radius a centered at (a, 0) in Cartesian coordinates. Similarly, ρ = 2a sin θ would be a circle centered at (0, a). So, comparing this to the given equation ρ = 4 cos θ, that should correspond to a circle with radius 2 (since 2a = 4 implies a = 2) centered at (2, 0). Let me verify that.To convert ρ = 4 cos θ to Cartesian coordinates, multiply both sides by ρ: ρ² = 4ρ cos θ. Since ρ² = x² + y² and ρ cos θ = x, this becomes x² + y² = 4x. Then, rearranging: x² - 4x + y² = 0. Completing the square for the x terms: x² - 4x + 4 + y² = 4 → (x - 2)² + y² = 4. Yes, so the circle has center (2, 0) and radius 2. Perfect, that checks out.So the center of circle C is at (2, 0). Now, I need to find the distance from this point (2, 0) to the line l, which has the equation x + y - 3 = 0.The formula for the distance from a point (x₀, y₀) to the line Ax + By + C = 0 is |Ax₀ + By₀ + C| / sqrt(A² + B²). So here, A = 1, B = 1, C = -3. The point is (2, 0). Plugging into the formula:Distance = |1*2 + 1*0 - 3| / sqrt(1² + 1²) = |2 + 0 - 3| / sqrt(2) = |-1| / sqrt(2) = 1 / sqrt(2). Which simplifies to sqrt(2)/2. Wait, let me double-check my steps. The line equation from the parametric equations: when I converted, I got y = -x + 3, so rearranged as x + y - 3 = 0. That's correct. So A = 1, B = 1, C = -3. Then, using the center (2, 0):Calculating numerator: 1*2 + 1*0 + (-3) = 2 + 0 - 3 = -1. Absolute value is 1. Denominator is sqrt(1 + 1) = sqrt(2). So yes, distance is 1/sqrt(2), which is equivalent to sqrt(2)/2. Rationalizing the denominator, it's often written as sqrt(2)/2. Either form is acceptable, but perhaps they want it in a simplified radical form. So sqrt(2)/2.Wait, but let me confirm once more. The formula is |Ax₀ + By₀ + C| / sqrt(A² + B²). So plugging in the numbers:A = 1, x₀ = 2, B = 1, y₀ = 0, C = -3.So |(1)(2) + (1)(0) + (-3)| = |2 + 0 - 3| = |-1| = 1.Denominator: sqrt(1² + 1²) = sqrt(2). So 1/sqrt(2) = sqrt(2)/2. Correct. So the distance is sqrt(2)/2.Hmm, seems straightforward. Let me just check if there's any chance I messed up the center of the circle. The polar equation was ρ = 4 cos θ. Conversion steps: multiplied by ρ to get ρ² = 4ρ cos θ, which is x² + y² = 4x. Then moved 4x to the left: x² - 4x + y² = 0. Completing the square for x: take half of -4, which is -2, square it, which is 4. So add 4 to both sides: (x² - 4x + 4) + y² = 4. So (x - 2)^2 + y^2 = 4. Therefore, center at (2, 0), radius 2. Correct. So center is (2, 0). The line is x + y - 3 = 0. Calculating distance as above. All steps seem correct.So, final answer should be sqrt(2)/2. Which is approximately 0.707, but exact value is sqrt(2)/2. Alternatively, if rationalized, but both forms are equivalent.Wait, but just to be thorough, maybe I should check using another method. For example, parametrically. Suppose I take a point on the line l and compute the vector from the center (2,0) to that point, then project onto the direction perpendicular to the line. But that might be more complicated. Alternatively, since I already converted the line to Cartesian form, and applied the formula, which is standard, I think that's solid.Alternatively, take two points on line l and compute the vector, find the direction, then compute the distance. Let's see. For instance, when t = 0, x = -1, y = 4. So point (-1, 4). When t = 1, x = 2*1 -1 = 1, y = 4 - 2*1 = 2. So another point (1, 2). The vector from (-1, 4) to (1, 2) is (2, -2), which is direction vector (2, -2) or simplified (1, -1). So the line has direction vector (1, -1). Therefore, the slope is -1, which matches the Cartesian equation y = -x + 3. Correct. The normal vector to the line would be perpendicular to the direction vector, so (1, 1), since the dot product with (1, -1) is 1*1 + 1*(-1) = 0. So the normal vector is (1, 1), which corresponds to the coefficients A and B in the standard line equation. So the distance formula using (1, 1) as normal vector is consistent.So, center at (2,0). The vector from any point on the line to the center is, say, from point (1,2) to center (2,0) is (1, -2). The distance is the absolute value of the scalar projection onto the unit normal vector. The unit normal vector is (1, 1)/sqrt(2). Then the scalar projection of (1, -2) onto this unit vector is (1*1 + (-2)*1)/sqrt(2) = (1 - 2)/sqrt(2) = (-1)/sqrt(2). Absolute value is 1/sqrt(2) = sqrt(2)/2. Same result. So this confirms the distance is sqrt(2)/2.Therefore, I'm confident the answer is sqrt(2)/2.Final AnswerThe distance from the center of circle ( C ) to line ( l ) is boxed{dfrac{sqrt{2}}{2}}.</think>To find the distance from the center of circle ( C ) to the line ( l ), we start by converting the given parametric equations of the line ( l ) into a Cartesian equation. The parametric equations are:[begin{cases}x = 2t - 1 y = 4 - 2tend{cases} quad (t in mathbb{R})]Solving for ( t ) in the ( x )-equation gives ( t = frac{x + 1}{2} ). Substituting this into the ( y )-equation:[y = 4 - 2 left( frac{x + 1}{2} right) = 4 - (x + 1) = 3 - x]Thus, the Cartesian equation of line ( l ) is ( x + y - 3 = 0 ).Next, we convert the polar equation of circle ( C ), ( rho = 4 cos theta ), to Cartesian coordinates. Multiplying both sides by ( rho ) gives:[rho^2 = 4 rho cos theta implies x^2 + y^2 = 4x]Rearranging and completing the square:[x^2 - 4x + y^2 = 0 implies (x - 2)^2 + y^2 = 4]The center of the circle is ( (2, 0) ).Using the distance formula from a point to a line, the distance from ( (2, 0) ) to the line ( x + y - 3 = 0 ) is:[frac{|1 cdot 2 + 1 cdot 0 - 3|}{sqrt{1^2 + 1^2}} = frac{|2 - 3|}{sqrt{2}} = frac{1}{sqrt{2}} = frac{sqrt{2}}{2}]The distance from the center of circle ( C ) to line ( l ) is boxed{dfrac{sqrt{2}}{2}}.,<think>Okay, let's see. I need to find the distance from the center of circle C to the line l. The circle C has a polar equation ρ = 4 cos θ, and the line l is given by parametric equations in Cartesian coordinates. Hmm. First, I should recall how to convert between polar and Cartesian coordinates, especially for circles. So, the polar equation ρ = 4 cos θ. I remember that in general, a polar equation of the form ρ = 2a cos θ represents a circle with radius a and center at (a, 0) in Cartesian coordinates. Let me verify that. If ρ = 2a cos θ, then multiplying both sides by ρ gives ρ² = 2a ρ cos θ. Converting to Cartesian coordinates, that's x² + y² = 2a x. Then rearranging, x² - 2a x + y² = 0, which can be written as (x - a)² + y² = a². So yes, that's a circle with center (a, 0) and radius a. In our case, the equation is ρ = 4 cos θ, so comparing to ρ = 2a cos θ, we see that 2a = 4, so a = 2. Therefore, the circle C has center at (2, 0) and radius 2. Got that. So the center of the circle is at (2, 0). Now, the line l is given parametrically as:x = 2t - 1y = 4 - 2tfor t ∈ ℝ. I need to find the distance from the center (2, 0) to this line l. To do that, I should first find the Cartesian equation of line l. Let me eliminate the parameter t. From the parametric equations:x = 2t - 1 → solve for t: t = (x + 1)/2Similarly, y = 4 - 2t → substitute t from above: y = 4 - 2*( (x + 1)/2 ) = 4 - (x + 1) = 4 - x -1 = 3 - xTherefore, the Cartesian equation of line l is y = -x + 3. Alternatively, written as x + y = 3. Alternatively, in standard form, it's x + y - 3 = 0. Now, the formula for the distance from a point (x₀, y₀) to the line Ax + By + C = 0 is |A x₀ + B y₀ + C| / sqrt(A² + B²). So in this case, the line l is x + y - 3 = 0, so A = 1, B = 1, C = -3. The center of the circle is (2, 0). Plugging into the formula: |1*2 + 1*0 - 3| / sqrt(1² + 1²) = |2 + 0 - 3| / sqrt(2) = |-1| / sqrt(2) = 1 / sqrt(2). But wait, rationalizing the denominator, that's sqrt(2)/2. So the distance is sqrt(2)/2. Let me double-check my steps. First, converting the parametric equations to Cartesian: correct. From x = 2t -1 and y = 4 - 2t, solving for t gives t = (x +1)/2, substituting into y gives y = 4 - 2*(x +1)/2 = 4 - (x +1) = 3 - x. So y = -x +3, correct. Then, writing the line in standard form: x + y - 3 = 0. Correct. Then, the center of the circle is (2,0). So using distance formula: |1*2 + 1*0 -3| / sqrt(1 +1) = |2 -3| / sqrt(2) = 1 / sqrt(2) = sqrt(2)/2. Yes, that seems right. Alternatively, if I use another method, like finding a point on the line and computing the vector projection, but that should give the same result. Alternatively, since the line is y = -x +3, another way to compute the distance is by taking a general point (x, y) on the line, and minimizing the distance squared to (2,0). But that would involve calculus, taking derivatives. But since the line is straight, the formula should be straightforward. Alternatively, let me recall that the distance from a point to a line can also be calculated using the formula: if the line is ax + by + c =0, then distance is |ax₀ + by₀ + c| / sqrt(a² + b²). Which is exactly what I used. So I think that's correct. But let me check with coordinates. Suppose I take a point on the line, say when t=0, x = -1, y = 4. Then the point (-1,4) is on line l. Then another point when t=1, x=2*1 -1=1, y=4-2*1=2. So point (1,2) is on line l. Then the vector from (2,0) to (1,2) is (-1, 2). The direction vector of line l can be found from parametric equations. Since x = 2t -1, y=4 -2t, so the direction vector is (2, -2). So the direction vector is (2, -2), which can be simplified to (1, -1). Therefore, the line l has direction vector (1, -1), so its slope is -1, which matches the equation y = -x +3. To compute the distance from (2,0) to line l, the formula using vectors: take a point P on the line, say (1,2), then the vector from P to the center (2,0) is (2 -1, 0 -2) = (1, -2). The distance is the magnitude of the component of this vector perpendicular to the line. The direction vector of the line is (1, -1). The unit normal vector to the line is (1,1)/sqrt(2), since the line's normal vector is (1,1) from the standard form equation. Therefore, the distance is the absolute value of the dot product of vector (1, -2) with (1,1)/sqrt(2). Compute (1*1 + (-2)*1)/sqrt(2) = (1 -2)/sqrt(2) = (-1)/sqrt(2). Absolute value is 1/sqrt(2), same as before. So sqrt(2)/2. Therefore, both methods confirm that the distance is sqrt(2)/2. But just to be thorough, let me use another point on the line. Take t=2: x=2*2 -1=3, y=4-2*2=0. So the point (3,0) is on the line l. Then the vector from (2,0) to (3,0) is (1,0). The unit normal vector is (1,1)/sqrt(2). The dot product is (1*1 + 0*1)/sqrt(2) = 1/sqrt(2). Absolute value is same as before. So again, 1/sqrt(2). Hmm, but wait, that's different. Wait, no. Wait, the distance should be the same regardless of the point we choose. Wait, but in this case, the point (3,0) is on the line l. So the vector from (2,0) to (3,0) is (1,0). The component of this vector in the direction of the normal is (1,0) ⋅ (1,1)/sqrt(2) = (1*1 + 0*1)/sqrt(2) = 1/sqrt(2). So same answer. So yeah, regardless of the point chosen on the line, the distance is sqrt(2)/2. Therefore, the distance from the center (2,0) to line l is sqrt(2)/2. But just to confirm, let me compute it using the standard formula again. Line equation: x + y -3 =0. Point (2,0). Plug in: |2 + 0 -3| / sqrt(1 +1) = | -1 | / sqrt(2) = 1/sqrt(2) = sqrt(2)/2. Yep. Therefore, the answer is sqrt(2)/2, which is approximately 0.707. But in exact terms, it's sqrt(2)/2. Wait, but maybe there's a mistake in converting the polar equation to Cartesian? Let me check again. Given ρ = 4 cos θ. Multiply both sides by ρ: ρ² = 4ρ cos θ. Then, replacing with Cartesian coordinates: x² + y² = 4x. Then rearranged: x² -4x + y² =0. Completing the square: x² -4x +4 + y² =4 → (x -2)² + y² =4. So center at (2,0), radius 2. Correct. So that's correct. So no mistake there. Therefore, conclusion: the distance is sqrt(2)/2. So the final answer is boxed{dfrac{sqrt{2}}{2}}.Final AnswerThe distance from the center of circle ( C ) to line ( l ) is boxed{dfrac{sqrt{2}}{2}}.</think>To find the distance from the center of circle ( C ) to line ( l ):1. Convert the polar equation of the circle to Cartesian coordinates: - The polar equation ( rho = 4 cos theta ) represents a circle. Converting this to Cartesian coordinates: [ rho = 4 cos theta implies rho^2 = 4 rho cos theta implies x^2 + y^2 = 4x ] Completing the square: [ x^2 - 4x + y^2 = 0 implies (x - 2)^2 + y^2 = 4 ] - The center of the circle ( C ) is ( (2, 0) ).2. Convert the parametric equations of line ( l ) to Cartesian form: - Parametric equations: [ begin{cases} x = 2t - 1 y = 4 - 2t end{cases} ] - Solving for ( t ) from the first equation: ( t = frac{x + 1}{2} ) - Substituting ( t ) into the second equation: [ y = 4 - 2 left( frac{x + 1}{2} right) = 4 - (x + 1) = 3 - x ] - The Cartesian equation of line ( l ) is ( x + y - 3 = 0 ).3. Calculate the distance from the center ( (2, 0) ) to the line ( x + y - 3 = 0 ): - Using the distance formula from a point to a line: [ text{Distance} = frac{|Ax_0 + By_0 + C|}{sqrt{A^2 + B^2}} ] - Here, ( A = 1 ), ( B = 1 ), ( C = -3 ), ( x_0 = 2 ), ( y_0 = 0 ): [ text{Distance} = frac{|1 cdot 2 + 1 cdot 0 - 3|}{sqrt{1^2 + 1^2}} = frac{|2 - 3|}{sqrt{2}} = frac{1}{sqrt{2}} = frac{sqrt{2}}{2} ]The distance from the center of circle ( C ) to line ( l ) is boxed{dfrac{sqrt{2}}{2}}.]

🔑:Given the parametric equations of line l:begin{cases}x = 2t - 1 y = 4 - 2tend{cases} quad (t in mathbb{R}),we can rewrite this as x + y - 3 = 0 by eliminating the parameter t.Since the polar equation of circle C is rho = 4costheta, squaring both sides gives:rho^2 = 4rhocostheta.This can be expressed in Cartesian coordinates as:x^2 + y^2 = 4x.Completing the square for x in x^2 - 4x, we get (x - 2)^2 + y^2 = 4.This shows that the center of the circle is at (2, 0).Now, to find the distance from the center of circle C to the line l, we use the formula for the distance from a point to a line in the plane, which is frac{|ax_1 + by_1 + c|}{sqrt{a^2 + b^2}} for a line ax + by + c = 0 and a point (x_1,y_1).The distance from the center of C to l is given by:frac{|2 + 0 - 3|}{sqrt{1^2 + 1^2}} = frac{|-1|}{sqrt{2}} = frac{sqrt{2}}{2}.Therefore, the answer is boxed{frac{sqrt{2}}{2}}.The solution involved converting the parameter equation and the polar equation to the standard Cartesian equation, finding the coordinates of the circle's center, and then using the point-to-line distance formula to find the distance from the center of circle C to line l. This problem tests the method of converting parametric equations and polar equations to standard equations, as well as the application of the distance formula, demonstrating a mathematical idea of transformation.

🧠:[<think>Okay, let me see. The question is asking which of the given square roots simplifies to 3√2. The options are A: √8, B: √9, C: √12, and D: √18. Hmm. Alright, so I need to simplify each of these square roots and see which one gives 3√2. Let me start with each option one by one.Starting with option A: √8. Let me try to simplify that. 8 can be broken down into factors where one of them is a perfect square. The factors of 8 are 4 and 2. Since 4 is a perfect square, I can write √8 as √(4*2). Then, using the property that √(a*b) = √a * √b, that becomes √4 * √2. √4 is 2, so this simplifies to 2√2. Hmm, okay, so √8 simplifies to 2√2. But the question is asking for 3√2, so A is not the correct answer. Let's cross that off.Next is option B: √9. Well, 9 is a perfect square. √9 is just 3. That's straightforward. So B simplifies to 3, which is different from 3√2. So B is not the answer either.Moving on to option C: √12. Let's break that down. 12 can be factored into 4 and 3. 4 is a perfect square. So √12 becomes √(4*3) = √4 * √3 = 2√3. That's 2 times the square root of 3, which is not the same as 3√2. So option C is out.Finally, option D: √18. Let me simplify that. 18 can be broken into 9 and 2. 9 is a perfect square. So √18 is √(9*2) = √9 * √2 = 3√2. Oh, that's exactly what we're looking for! So D simplifies to 3√2.Wait, let me double-check each step to make sure I didn't make a mistake. Starting with A: 8 = 4*2, √4 is 2, so 2√2. Correct. B: √9 is 3. Correct. C: 12 = 4*3, √4 is 2, so 2√3. Correct. D: 18 = 9*2, √9 is 3, so 3√2. Yep, that's right. So D is the correct answer.Just to ensure there's no other way any of these could simplify to 3√2. Let me consider. Maybe if someone factors differently. For example, with √18, if you factor 18 as 3*6, then √(3*6) = √3 * √6. But √6 isn't a perfect square, so that doesn't help. But the correct way is to factor out the largest perfect square, which is 9. So that gives 3√2. Right.Similarly, if someone tried to factor 12 as 2*6, but neither 2 nor 6 are perfect squares. So that's not helpful. So yeah, the process is correct. So I think D is definitely the answer here.Final Answerboxed{D}</think>To determine which square root simplifies to (3sqrt{2}), we evaluate each option:- Option A: (sqrt{8}) Factor 8 into (4 times 2): [ sqrt{8} = sqrt{4 times 2} = sqrt{4} times sqrt{2} = 2sqrt{2} ] This simplifies to (2sqrt{2}), not (3sqrt{2}).- Option B: (sqrt{9}) Since 9 is a perfect square: [ sqrt{9} = 3 ] This simplifies to 3, not (3sqrt{2}).- Option C: (sqrt{12}) Factor 12 into (4 times 3): [ sqrt{12} = sqrt{4 times 3} = sqrt{4} times sqrt{3} = 2sqrt{3} ] This simplifies to (2sqrt{3}), not (3sqrt{2}).- Option D: (sqrt{18}) Factor 18 into (9 times 2): [ sqrt{18} = sqrt{9 times 2} = sqrt{9} times sqrt{2} = 3sqrt{2} ] This simplifies to (3sqrt{2}), which matches the desired result.Thus, the correct answer is (boxed{D}).,<think>Okay, let's see here. The question is asking which of the given square roots simplifies to 3√2. The options are A: √8, B: √9, C: √12, and D: √18. Hmm, I need to figure out each one step by step.First, I remember that simplifying a square root involves factoring out perfect squares. A perfect square is a number that's the square of an integer, like 4, 9, 16, etc. So, for each option, I should break down the number under the square root into factors where at least one is a perfect square. Then, take the square root of that perfect square and leave the remaining under the radical.Starting with option A: √8. Let me factor 8. The factors of 8 are 4 and 2 because 4*2=8. And 4 is a perfect square since 2²=4. So, √8 can be written as √(4*2). Then, using the property that √(a*b) = √a * √b, this becomes √4 * √2. √4 is 2, so this simplifies to 2√2. But the question is looking for 3√2, so A simplifies to 2√2, which isn't the one we need. So A is out.Next, option B: √9. That's straightforward. Since 9 is a perfect square (3²=9), √9 is just 3. There's no radical part here. The problem wants 3√2, which is 3 multiplied by the square root of 2. But option B is just 3, so that's not it either. So B is not the answer.Moving on to option C: √12. Let's factor 12. The factors here could be 4 and 3 because 4*3=12, and 4 is a perfect square. So, √12 = √(4*3) = √4 * √3 = 2√3. Again, this gives 2√3, which is different from 3√2. The coefficient is 2 instead of 3, and the radical is √3 instead of √2. So C is not the correct choice either.Now, option D: √18. Let's factor 18. The factors could be 9 and 2, since 9*2=18, and 9 is a perfect square (3²=9). Therefore, √18 = √(9*2) = √9 * √2 = 3√2. That matches exactly what the question is asking for: 3 multiplied by √2. So D simplifies to 3√2.Wait, let me double-check each step to be sure I didn't make a mistake. For A: 8=4*2, sqrt(4)=2, so 2√2. Correct. B: sqrt(9)=3. Right. C: 12=4*3, sqrt(4)=2, so 2√3. Yep. D: 18=9*2, sqrt(9)=3, so 3√2. All steps check out. So D is the correct answer.Just to make sure there isn't another possible factorization for any of the options that might give a different result. For example, if I took 18 and factored it as 6*3, but neither 6 nor 3 are perfect squares. So that wouldn't help. Similarly, 8 could be 2*4, but 4 is already the perfect square. So no, my initial approach was correct. The key is to factor out the largest perfect square possible. So in each case, the largest perfect square factor is what's important.Therefore, the answer is D: √18, which simplifies to 3√2.Final Answerboxed{D}</think>To determine which square root simplifies to (3sqrt{2}), we need to factor each option into a product of a perfect square and another number, then take the square root of the perfect square.- Option A: (sqrt{8}) Factor 8 as (4 times 2): [ sqrt{8} = sqrt{4 times 2} = sqrt{4} times sqrt{2} = 2sqrt{2} ] This simplifies to (2sqrt{2}), not (3sqrt{2}).- Option B: (sqrt{9}) Since 9 is a perfect square: [ sqrt{9} = 3 ] This simplifies to 3, not (3sqrt{2}).- Option C: (sqrt{12}) Factor 12 as (4 times 3): [ sqrt{12} = sqrt{4 times 3} = sqrt{4} times sqrt{3} = 2sqrt{3} ] This simplifies to (2sqrt{3}), not (3sqrt{2}).- Option D: (sqrt{18}) Factor 18 as (9 times 2): [ sqrt{18} = sqrt{9 times 2} = sqrt{9} times sqrt{2} = 3sqrt{2} ] This simplifies to (3sqrt{2}), which matches the desired form.Thus, the correct answer is (boxed{D}).]

🔑:To find which square root simplifies to 3sqrt{2}, we simplify each option:- For option A: sqrt{8} = sqrt{4 cdot 2} = sqrt{4}sqrt{2} = 2sqrt{2}. This does not equal 3sqrt{2}.- For option B: sqrt{9} = 3. This is because 9 is a perfect square, and its square root is the integer 3, not in the form of 3sqrt{2}.- For option C: sqrt{12} = sqrt{4 cdot 3} = sqrt{4}sqrt{3} = 2sqrt{3}. This does not equal 3sqrt{2}.- For option D: sqrt{18} = sqrt{9 cdot 2} = sqrt{9}sqrt{2} = 3sqrt{2}. This matches the requirement of being equal to 3sqrt{2}.Therefore, the correct answer is option D, which simplifies to 3sqrt{2}.[boxed{D}]