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Simplifying (5 + w)(w + -2) = 162 Reorder the terms: (5 + w)(-2 + w) = 162 Multiply (5 + w) * (-2 + w) (5(-2 + w) + w(-2 + w)) = 162 ((-2 * 5 + w * 5) + w(-2 + w)) = 162 ((-10 + 5w) + w(-2 + w)) = 162 (-10 + 5w + (-2 * w + w * w)) = 162 (-10 + 5w + (-2w + w2)) = 162 Combine like terms: 5w + -2w = 3w (-10 + 3w + w2) = 162 Solving -10 + 3w + w2 = 162 Solving for variable 'w'. Reorder the terms: -10 + -162 + 3w + w2 = 162 + -162 Combine like terms: -10 + -162 = -172 -172 + 3w + w2 = 162 + -162 Combine like terms: 162 + -162 = 0 -172 + 3w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '172' to each side of the equation. -172 + 3w + 172 + w2 = 0 + 172 Reorder the terms: -172 + 172 + 3w + w2 = 0 + 172 Combine like terms: -172 + 172 = 0 0 + 3w + w2 = 0 + 172 3w + w2 = 0 + 172 Combine like terms: 0 + 172 = 172 3w + w2 = 172 The w term is 3w. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3w + 2.25 + w2 = 172 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 172 + 2.25 Combine like terms: 172 + 2.25 = 174.25 2.25 + 3w + w2 = 174.25 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 174.25 Calculate the square root of the right side: 13.200378782 Break this problem into two subproblems by setting (w + 1.5) equal to 13.200378782 and -13.200378782.Subproblem 1
w + 1.5 = 13.200378782 Simplifying w + 1.5 = 13.200378782 Reorder the terms: 1.5 + w = 13.200378782 Solving 1.5 + w = 13.200378782 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 13.200378782 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 13.200378782 + -1.5 w = 13.200378782 + -1.5 Combine like terms: 13.200378782 + -1.5 = 11.700378782 w = 11.700378782 Simplifying w = 11.700378782Subproblem 2
w + 1.5 = -13.200378782 Simplifying w + 1.5 = -13.200378782 Reorder the terms: 1.5 + w = -13.200378782 Solving 1.5 + w = -13.200378782 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -13.200378782 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -13.200378782 + -1.5 w = -13.200378782 + -1.5 Combine like terms: -13.200378782 + -1.5 = -14.700378782 w = -14.700378782 Simplifying w = -14.700378782Solution
The solution to the problem is based on the solutions from the subproblems. w = {11.700378782, -14.700378782}
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