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Simplifying (w + 4)(2w + -5 + 4) = 132 Reorder the terms: (4 + w)(2w + -5 + 4) = 132 Reorder the terms: (4 + w)(-5 + 4 + 2w) = 132 Combine like terms: -5 + 4 = -1 (4 + w)(-1 + 2w) = 132 Multiply (4 + w) * (-1 + 2w) (4(-1 + 2w) + w(-1 + 2w)) = 132 ((-1 * 4 + 2w * 4) + w(-1 + 2w)) = 132 ((-4 + 8w) + w(-1 + 2w)) = 132 (-4 + 8w + (-1 * w + 2w * w)) = 132 (-4 + 8w + (-1w + 2w2)) = 132 Combine like terms: 8w + -1w = 7w (-4 + 7w + 2w2) = 132 Solving -4 + 7w + 2w2 = 132 Solving for variable 'w'. Reorder the terms: -4 + -132 + 7w + 2w2 = 132 + -132 Combine like terms: -4 + -132 = -136 -136 + 7w + 2w2 = 132 + -132 Combine like terms: 132 + -132 = 0 -136 + 7w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -68 + 3.5w + w2 = 0 Move the constant term to the right: Add '68' to each side of the equation. -68 + 3.5w + 68 + w2 = 0 + 68 Reorder the terms: -68 + 68 + 3.5w + w2 = 0 + 68 Combine like terms: -68 + 68 = 0 0 + 3.5w + w2 = 0 + 68 3.5w + w2 = 0 + 68 Combine like terms: 0 + 68 = 68 3.5w + w2 = 68 The w term is 3.5w. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5w + 3.0625 + w2 = 68 + 3.0625 Reorder the terms: 3.0625 + 3.5w + w2 = 68 + 3.0625 Combine like terms: 68 + 3.0625 = 71.0625 3.0625 + 3.5w + w2 = 71.0625 Factor a perfect square on the left side: (w + 1.75)(w + 1.75) = 71.0625 Calculate the square root of the right side: 8.42985765 Break this problem into two subproblems by setting (w + 1.75) equal to 8.42985765 and -8.42985765.Subproblem 1
w + 1.75 = 8.42985765 Simplifying w + 1.75 = 8.42985765 Reorder the terms: 1.75 + w = 8.42985765 Solving 1.75 + w = 8.42985765 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + w = 8.42985765 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + w = 8.42985765 + -1.75 w = 8.42985765 + -1.75 Combine like terms: 8.42985765 + -1.75 = 6.67985765 w = 6.67985765 Simplifying w = 6.67985765Subproblem 2
w + 1.75 = -8.42985765 Simplifying w + 1.75 = -8.42985765 Reorder the terms: 1.75 + w = -8.42985765 Solving 1.75 + w = -8.42985765 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + w = -8.42985765 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + w = -8.42985765 + -1.75 w = -8.42985765 + -1.75 Combine like terms: -8.42985765 + -1.75 = -10.17985765 w = -10.17985765 Simplifying w = -10.17985765Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.67985765, -10.17985765}
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