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Simplifying 2(2w + 5)(w) = 259 Reorder the terms: 2(5 + 2w)(w) = 259 Reorder the terms for easier multiplication: 2w(5 + 2w) = 259 (5 * 2w + 2w * 2w) = 259 (10w + 4w2) = 259 Solving 10w + 4w2 = 259 Solving for variable 'w'. Reorder the terms: -259 + 10w + 4w2 = 259 + -259 Combine like terms: 259 + -259 = 0 -259 + 10w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -64.75 + 2.5w + w2 = 0 Move the constant term to the right: Add '64.75' to each side of the equation. -64.75 + 2.5w + 64.75 + w2 = 0 + 64.75 Reorder the terms: -64.75 + 64.75 + 2.5w + w2 = 0 + 64.75 Combine like terms: -64.75 + 64.75 = 0.00 0.00 + 2.5w + w2 = 0 + 64.75 2.5w + w2 = 0 + 64.75 Combine like terms: 0 + 64.75 = 64.75 2.5w + w2 = 64.75 The w term is 2.5w. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5w + 1.5625 + w2 = 64.75 + 1.5625 Reorder the terms: 1.5625 + 2.5w + w2 = 64.75 + 1.5625 Combine like terms: 64.75 + 1.5625 = 66.3125 1.5625 + 2.5w + w2 = 66.3125 Factor a perfect square on the left side: (w + 1.25)(w + 1.25) = 66.3125 Calculate the square root of the right side: 8.143248737 Break this problem into two subproblems by setting (w + 1.25) equal to 8.143248737 and -8.143248737.Subproblem 1
w + 1.25 = 8.143248737 Simplifying w + 1.25 = 8.143248737 Reorder the terms: 1.25 + w = 8.143248737 Solving 1.25 + w = 8.143248737 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + w = 8.143248737 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + w = 8.143248737 + -1.25 w = 8.143248737 + -1.25 Combine like terms: 8.143248737 + -1.25 = 6.893248737 w = 6.893248737 Simplifying w = 6.893248737Subproblem 2
w + 1.25 = -8.143248737 Simplifying w + 1.25 = -8.143248737 Reorder the terms: 1.25 + w = -8.143248737 Solving 1.25 + w = -8.143248737 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + w = -8.143248737 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + w = -8.143248737 + -1.25 w = -8.143248737 + -1.25 Combine like terms: -8.143248737 + -1.25 = -9.393248737 w = -9.393248737 Simplifying w = -9.393248737Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.893248737, -9.393248737}
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