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Simplifying 32w2 + 4w + -4 = 0 Reorder the terms: -4 + 4w + 32w2 = 0 Solving -4 + 4w + 32w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '4'. 4(-1 + w + 8w2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-1 + w + 8w2)' equal to zero and attempt to solve: Simplifying -1 + w + 8w2 = 0 Solving -1 + w + 8w2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + 0.125w + w2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + 0.125w + 0.125 + w2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + 0.125w + w2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + 0.125w + w2 = 0 + 0.125 0.125w + w2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 0.125w + w2 = 0.125 The w term is w. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.125w + 0.25 + w2 = 0.125 + 0.25 Reorder the terms: 0.25 + 0.125w + w2 = 0.125 + 0.25 Combine like terms: 0.125 + 0.25 = 0.375 0.25 + 0.125w + w2 = 0.375 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 0.375 Calculate the square root of the right side: 0.612372436 Break this problem into two subproblems by setting (w + 0.5) equal to 0.612372436 and -0.612372436.Subproblem 1
w + 0.5 = 0.612372436 Simplifying w + 0.5 = 0.612372436 Reorder the terms: 0.5 + w = 0.612372436 Solving 0.5 + w = 0.612372436 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 0.612372436 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 0.612372436 + -0.5 w = 0.612372436 + -0.5 Combine like terms: 0.612372436 + -0.5 = 0.112372436 w = 0.112372436 Simplifying w = 0.112372436Subproblem 2
w + 0.5 = -0.612372436 Simplifying w + 0.5 = -0.612372436 Reorder the terms: 0.5 + w = -0.612372436 Solving 0.5 + w = -0.612372436 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -0.612372436 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -0.612372436 + -0.5 w = -0.612372436 + -0.5 Combine like terms: -0.612372436 + -0.5 = -1.112372436 w = -1.112372436 Simplifying w = -1.112372436Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.112372436, -1.112372436}Solution
w = {0.112372436, -1.112372436}
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