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Simplifying 4w2 + 38w + -32 = 0 Reorder the terms: -32 + 38w + 4w2 = 0 Solving -32 + 38w + 4w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '2'. 2(-16 + 19w + 2w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-16 + 19w + 2w2)' equal to zero and attempt to solve: Simplifying -16 + 19w + 2w2 = 0 Solving -16 + 19w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -8 + 9.5w + w2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 9.5w + 8 + w2 = 0 + 8 Reorder the terms: -8 + 8 + 9.5w + w2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 9.5w + w2 = 0 + 8 9.5w + w2 = 0 + 8 Combine like terms: 0 + 8 = 8 9.5w + w2 = 8 The w term is 9.5w. Take half its coefficient (4.75). Square it (22.5625) and add it to both sides. Add '22.5625' to each side of the equation. 9.5w + 22.5625 + w2 = 8 + 22.5625 Reorder the terms: 22.5625 + 9.5w + w2 = 8 + 22.5625 Combine like terms: 8 + 22.5625 = 30.5625 22.5625 + 9.5w + w2 = 30.5625 Factor a perfect square on the left side: (w + 4.75)(w + 4.75) = 30.5625 Calculate the square root of the right side: 5.528336097 Break this problem into two subproblems by setting (w + 4.75) equal to 5.528336097 and -5.528336097.Subproblem 1
w + 4.75 = 5.528336097 Simplifying w + 4.75 = 5.528336097 Reorder the terms: 4.75 + w = 5.528336097 Solving 4.75 + w = 5.528336097 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + w = 5.528336097 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + w = 5.528336097 + -4.75 w = 5.528336097 + -4.75 Combine like terms: 5.528336097 + -4.75 = 0.778336097 w = 0.778336097 Simplifying w = 0.778336097Subproblem 2
w + 4.75 = -5.528336097 Simplifying w + 4.75 = -5.528336097 Reorder the terms: 4.75 + w = -5.528336097 Solving 4.75 + w = -5.528336097 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + w = -5.528336097 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + w = -5.528336097 + -4.75 w = -5.528336097 + -4.75 Combine like terms: -5.528336097 + -4.75 = -10.278336097 w = -10.278336097 Simplifying w = -10.278336097Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.778336097, -10.278336097}Solution
w = {0.778336097, -10.278336097}
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