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Simplifying 2w(2w + -23) = -3w + 10 Reorder the terms: 2w(-23 + 2w) = -3w + 10 (-23 * 2w + 2w * 2w) = -3w + 10 (-46w + 4w2) = -3w + 10 Reorder the terms: -46w + 4w2 = 10 + -3w Solving -46w + 4w2 = 10 + -3w Solving for variable 'w'. Reorder the terms: -10 + -46w + 3w + 4w2 = 10 + -3w + -10 + 3w Combine like terms: -46w + 3w = -43w -10 + -43w + 4w2 = 10 + -3w + -10 + 3w Reorder the terms: -10 + -43w + 4w2 = 10 + -10 + -3w + 3w Combine like terms: 10 + -10 = 0 -10 + -43w + 4w2 = 0 + -3w + 3w -10 + -43w + 4w2 = -3w + 3w Combine like terms: -3w + 3w = 0 -10 + -43w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.5 + -10.75w + w2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -10.75w + 2.5 + w2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -10.75w + w2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -10.75w + w2 = 0 + 2.5 -10.75w + w2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -10.75w + w2 = 2.5 The w term is -10.75w. Take half its coefficient (-5.375). Square it (28.890625) and add it to both sides. Add '28.890625' to each side of the equation. -10.75w + 28.890625 + w2 = 2.5 + 28.890625 Reorder the terms: 28.890625 + -10.75w + w2 = 2.5 + 28.890625 Combine like terms: 2.5 + 28.890625 = 31.390625 28.890625 + -10.75w + w2 = 31.390625 Factor a perfect square on the left side: (w + -5.375)(w + -5.375) = 31.390625 Calculate the square root of the right side: 5.602733708 Break this problem into two subproblems by setting (w + -5.375) equal to 5.602733708 and -5.602733708.Subproblem 1
w + -5.375 = 5.602733708 Simplifying w + -5.375 = 5.602733708 Reorder the terms: -5.375 + w = 5.602733708 Solving -5.375 + w = 5.602733708 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '5.375' to each side of the equation. -5.375 + 5.375 + w = 5.602733708 + 5.375 Combine like terms: -5.375 + 5.375 = 0.000 0.000 + w = 5.602733708 + 5.375 w = 5.602733708 + 5.375 Combine like terms: 5.602733708 + 5.375 = 10.977733708 w = 10.977733708 Simplifying w = 10.977733708Subproblem 2
w + -5.375 = -5.602733708 Simplifying w + -5.375 = -5.602733708 Reorder the terms: -5.375 + w = -5.602733708 Solving -5.375 + w = -5.602733708 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '5.375' to each side of the equation. -5.375 + 5.375 + w = -5.602733708 + 5.375 Combine like terms: -5.375 + 5.375 = 0.000 0.000 + w = -5.602733708 + 5.375 w = -5.602733708 + 5.375 Combine like terms: -5.602733708 + 5.375 = -0.227733708 w = -0.227733708 Simplifying w = -0.227733708Solution
The solution to the problem is based on the solutions from the subproblems. w = {10.977733708, -0.227733708}
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