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Simplifying 6w2 + 6w + -31 = 0 Reorder the terms: -31 + 6w + 6w2 = 0 Solving -31 + 6w + 6w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -5.166666667 + w + w2 = 0 Move the constant term to the right: Add '5.166666667' to each side of the equation. -5.166666667 + w + 5.166666667 + w2 = 0 + 5.166666667 Reorder the terms: -5.166666667 + 5.166666667 + w + w2 = 0 + 5.166666667 Combine like terms: -5.166666667 + 5.166666667 = 0.000000000 0.000000000 + w + w2 = 0 + 5.166666667 w + w2 = 0 + 5.166666667 Combine like terms: 0 + 5.166666667 = 5.166666667 w + w2 = 5.166666667 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 = 5.166666667 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + w2 = 5.166666667 + 0.25 Combine like terms: 5.166666667 + 0.25 = 5.416666667 1.25 + w2 = 5.416666667 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 5.416666667 Calculate the square root of the right side: 2.327373341 Break this problem into two subproblems by setting (w + 0.5) equal to 2.327373341 and -2.327373341.Subproblem 1
w + 0.5 = 2.327373341 Simplifying w + 0.5 = 2.327373341 Reorder the terms: 0.5 + w = 2.327373341 Solving 0.5 + w = 2.327373341 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 = 2.327373341 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 2.327373341 + -0.5 w = 2.327373341 + -0.5 Combine like terms: 2.327373341 + -0.5 = 1.827373341 w = 1.827373341 Simplifying w = 1.827373341Subproblem 2
w + 0.5 = -2.327373341 Simplifying w + 0.5 = -2.327373341 Reorder the terms: 0.5 + w = -2.327373341 Solving 0.5 + w = -2.327373341 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 = -2.327373341 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -2.327373341 + -0.5 w = -2.327373341 + -0.5 Combine like terms: -2.327373341 + -0.5 = -2.827373341 w = -2.827373341 Simplifying w = -2.827373341Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.827373341, -2.827373341}
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