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Simplifying 7 + -10w = w(1 + -1w) 7 + -10w = (1 * w + -1w * w) 7 + -10w = (1w + -1w2) Solving 7 + -10w = 1w + -1w2 Solving for variable 'w'. Combine like terms: -10w + -1w = -11w 7 + -11w + w2 = 1w + -1w2 + -1w + w2 Reorder the terms: 7 + -11w + w2 = 1w + -1w + -1w2 + w2 Combine like terms: 1w + -1w = 0 7 + -11w + w2 = 0 + -1w2 + w2 7 + -11w + w2 = -1w2 + w2 Combine like terms: -1w2 + w2 = 0 7 + -11w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '-7' to each side of the equation. 7 + -11w + -7 + w2 = 0 + -7 Reorder the terms: 7 + -7 + -11w + w2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + -11w + w2 = 0 + -7 -11w + w2 = 0 + -7 Combine like terms: 0 + -7 = -7 -11w + w2 = -7 The w term is -11w. Take half its coefficient (-5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. -11w + 30.25 + w2 = -7 + 30.25 Reorder the terms: 30.25 + -11w + w2 = -7 + 30.25 Combine like terms: -7 + 30.25 = 23.25 30.25 + -11w + w2 = 23.25 Factor a perfect square on the left side: (w + -5.5)(w + -5.5) = 23.25 Calculate the square root of the right side: 4.82182538 Break this problem into two subproblems by setting (w + -5.5) equal to 4.82182538 and -4.82182538.Subproblem 1
w + -5.5 = 4.82182538 Simplifying w + -5.5 = 4.82182538 Reorder the terms: -5.5 + w = 4.82182538 Solving -5.5 + w = 4.82182538 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + w = 4.82182538 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + w = 4.82182538 + 5.5 w = 4.82182538 + 5.5 Combine like terms: 4.82182538 + 5.5 = 10.32182538 w = 10.32182538 Simplifying w = 10.32182538Subproblem 2
w + -5.5 = -4.82182538 Simplifying w + -5.5 = -4.82182538 Reorder the terms: -5.5 + w = -4.82182538 Solving -5.5 + w = -4.82182538 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + w = -4.82182538 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + w = -4.82182538 + 5.5 w = -4.82182538 + 5.5 Combine like terms: -4.82182538 + 5.5 = 0.67817462 w = 0.67817462 Simplifying w = 0.67817462Solution
The solution to the problem is based on the solutions from the subproblems. w = {10.32182538, 0.67817462}
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