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Simplifying 2w2 + 2w + 3 = 70 Reorder the terms: 3 + 2w + 2w2 = 70 Solving 3 + 2w + 2w2 = 70 Solving for variable 'w'. Reorder the terms: 3 + -70 + 2w + 2w2 = 70 + -70 Combine like terms: 3 + -70 = -67 -67 + 2w + 2w2 = 70 + -70 Combine like terms: 70 + -70 = 0 -67 + 2w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -33.5 + w + w2 = 0 Move the constant term to the right: Add '33.5' to each side of the equation. -33.5 + w + 33.5 + w2 = 0 + 33.5 Reorder the terms: -33.5 + 33.5 + w + w2 = 0 + 33.5 Combine like terms: -33.5 + 33.5 = 0.0 0.0 + w + w2 = 0 + 33.5 w + w2 = 0 + 33.5 Combine like terms: 0 + 33.5 = 33.5 w + w2 = 33.5 The w term is w. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + w2 = 33.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + w2 = 33.5 + 0.25 Combine like terms: 33.5 + 0.25 = 33.75 1.25 + w2 = 33.75 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 33.75 Calculate the square root of the right side: 5.809475019 Break this problem into two subproblems by setting (w + 0.5) equal to 5.809475019 and -5.809475019.Subproblem 1
w + 0.5 = 5.809475019 Simplifying w + 0.5 = 5.809475019 Reorder the terms: 0.5 + w = 5.809475019 Solving 0.5 + w = 5.809475019 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 5.809475019 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 5.809475019 + -0.5 w = 5.809475019 + -0.5 Combine like terms: 5.809475019 + -0.5 = 5.309475019 w = 5.309475019 Simplifying w = 5.309475019Subproblem 2
w + 0.5 = -5.809475019 Simplifying w + 0.5 = -5.809475019 Reorder the terms: 0.5 + w = -5.809475019 Solving 0.5 + w = -5.809475019 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -5.809475019 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -5.809475019 + -0.5 w = -5.809475019 + -0.5 Combine like terms: -5.809475019 + -0.5 = -6.309475019 w = -6.309475019 Simplifying w = -6.309475019Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.309475019, -6.309475019}
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