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Simplifying 3w3 + -21w2 + 12w = 0 Reorder the terms: 12w + -21w2 + 3w3 = 0 Solving 12w + -21w2 + 3w3 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '3w'. 3w(4 + -7w + w2) = 0 Ignore the factor 3.Subproblem 1
Set the factor 'w' equal to zero and attempt to solve: Simplifying w = 0 Solving w = 0 Move all terms containing w to the left, all other terms to the right. Simplifying w = 0Subproblem 2
Set the factor '(4 + -7w + w2)' equal to zero and attempt to solve: Simplifying 4 + -7w + w2 = 0 Solving 4 + -7w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '-4' to each side of the equation. 4 + -7w + -4 + w2 = 0 + -4 Reorder the terms: 4 + -4 + -7w + w2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -7w + w2 = 0 + -4 -7w + w2 = 0 + -4 Combine like terms: 0 + -4 = -4 -7w + w2 = -4 The w term is -7w. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7w + 12.25 + w2 = -4 + 12.25 Reorder the terms: 12.25 + -7w + w2 = -4 + 12.25 Combine like terms: -4 + 12.25 = 8.25 12.25 + -7w + w2 = 8.25 Factor a perfect square on the left side: (w + -3.5)(w + -3.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (w + -3.5) equal to 2.872281323 and -2.872281323.Subproblem 1
w + -3.5 = 2.872281323 Simplifying w + -3.5 = 2.872281323 Reorder the terms: -3.5 + w = 2.872281323 Solving -3.5 + w = 2.872281323 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + w = 2.872281323 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + w = 2.872281323 + 3.5 w = 2.872281323 + 3.5 Combine like terms: 2.872281323 + 3.5 = 6.372281323 w = 6.372281323 Simplifying w = 6.372281323Subproblem 2
w + -3.5 = -2.872281323 Simplifying w + -3.5 = -2.872281323 Reorder the terms: -3.5 + w = -2.872281323 Solving -3.5 + w = -2.872281323 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + w = -2.872281323 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + w = -2.872281323 + 3.5 w = -2.872281323 + 3.5 Combine like terms: -2.872281323 + 3.5 = 0.627718677 w = 0.627718677 Simplifying w = 0.627718677Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.372281323, 0.627718677}Solution
w = {0, 6.372281323, 0.627718677}
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