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Simplifying (2w + -5) * w = 65 Reorder the terms: (-5 + 2w) * w = 65 Reorder the terms for easier multiplication: w(-5 + 2w) = 65 (-5 * w + 2w * w) = 65 (-5w + 2w2) = 65 Solving -5w + 2w2 = 65 Solving for variable 'w'. Reorder the terms: -65 + -5w + 2w2 = 65 + -65 Combine like terms: 65 + -65 = 0 -65 + -5w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -32.5 + -2.5w + w2 = 0 Move the constant term to the right: Add '32.5' to each side of the equation. -32.5 + -2.5w + 32.5 + w2 = 0 + 32.5 Reorder the terms: -32.5 + 32.5 + -2.5w + w2 = 0 + 32.5 Combine like terms: -32.5 + 32.5 = 0.0 0.0 + -2.5w + w2 = 0 + 32.5 -2.5w + w2 = 0 + 32.5 Combine like terms: 0 + 32.5 = 32.5 -2.5w + w2 = 32.5 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 = 32.5 + 1.5625 Reorder the terms: 1.5625 + -2.5w + w2 = 32.5 + 1.5625 Combine like terms: 32.5 + 1.5625 = 34.0625 1.5625 + -2.5w + w2 = 34.0625 Factor a perfect square on the left side: (w + -1.25)(w + -1.25) = 34.0625 Calculate the square root of the right side: 5.836308765 Break this problem into two subproblems by setting (w + -1.25) equal to 5.836308765 and -5.836308765.Subproblem 1
w + -1.25 = 5.836308765 Simplifying w + -1.25 = 5.836308765 Reorder the terms: -1.25 + w = 5.836308765 Solving -1.25 + w = 5.836308765 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 = 5.836308765 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + w = 5.836308765 + 1.25 w = 5.836308765 + 1.25 Combine like terms: 5.836308765 + 1.25 = 7.086308765 w = 7.086308765 Simplifying w = 7.086308765Subproblem 2
w + -1.25 = -5.836308765 Simplifying w + -1.25 = -5.836308765 Reorder the terms: -1.25 + w = -5.836308765 Solving -1.25 + w = -5.836308765 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 = -5.836308765 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + w = -5.836308765 + 1.25 w = -5.836308765 + 1.25 Combine like terms: -5.836308765 + 1.25 = -4.586308765 w = -4.586308765 Simplifying w = -4.586308765Solution
The solution to the problem is based on the solutions from the subproblems. w = {7.086308765, -4.586308765}
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