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Simplifying 15w2 + 31w + -28 = 0 Reorder the terms: -28 + 31w + 15w2 = 0 Solving -28 + 31w + 15w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -1.866666667 + 2.066666667w + w2 = 0 Move the constant term to the right: Add '1.866666667' to each side of the equation. -1.866666667 + 2.066666667w + 1.866666667 + w2 = 0 + 1.866666667 Reorder the terms: -1.866666667 + 1.866666667 + 2.066666667w + w2 = 0 + 1.866666667 Combine like terms: -1.866666667 + 1.866666667 = 0.000000000 0.000000000 + 2.066666667w + w2 = 0 + 1.866666667 2.066666667w + w2 = 0 + 1.866666667 Combine like terms: 0 + 1.866666667 = 1.866666667 2.066666667w + w2 = 1.866666667 The w term is 2.066666667w. Take half its coefficient (1.033333334). Square it (1.067777779) and add it to both sides. Add '1.067777779' to each side of the equation. 2.066666667w + 1.067777779 + w2 = 1.866666667 + 1.067777779 Reorder the terms: 1.067777779 + 2.066666667w + w2 = 1.866666667 + 1.067777779 Combine like terms: 1.866666667 + 1.067777779 = 2.934444446 1.067777779 + 2.066666667w + w2 = 2.934444446 Factor a perfect square on the left side: (w + 1.033333334)(w + 1.033333334) = 2.934444446 Calculate the square root of the right side: 1.713022021 Break this problem into two subproblems by setting (w + 1.033333334) equal to 1.713022021 and -1.713022021.Subproblem 1
w + 1.033333334 = 1.713022021 Simplifying w + 1.033333334 = 1.713022021 Reorder the terms: 1.033333334 + w = 1.713022021 Solving 1.033333334 + w = 1.713022021 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.033333334' to each side of the equation. 1.033333334 + -1.033333334 + w = 1.713022021 + -1.033333334 Combine like terms: 1.033333334 + -1.033333334 = 0.000000000 0.000000000 + w = 1.713022021 + -1.033333334 w = 1.713022021 + -1.033333334 Combine like terms: 1.713022021 + -1.033333334 = 0.679688687 w = 0.679688687 Simplifying w = 0.679688687Subproblem 2
w + 1.033333334 = -1.713022021 Simplifying w + 1.033333334 = -1.713022021 Reorder the terms: 1.033333334 + w = -1.713022021 Solving 1.033333334 + w = -1.713022021 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.033333334' to each side of the equation. 1.033333334 + -1.033333334 + w = -1.713022021 + -1.033333334 Combine like terms: 1.033333334 + -1.033333334 = 0.000000000 0.000000000 + w = -1.713022021 + -1.033333334 w = -1.713022021 + -1.033333334 Combine like terms: -1.713022021 + -1.033333334 = -2.746355355 w = -2.746355355 Simplifying w = -2.746355355Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.679688687, -2.746355355}
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