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Simplifying 8w(w + -1) = (3w + 7) + 2w Reorder the terms: 8w(-1 + w) = (3w + 7) + 2w (-1 * 8w + w * 8w) = (3w + 7) + 2w (-8w + 8w2) = (3w + 7) + 2w Reorder the terms: -8w + 8w2 = (7 + 3w) + 2w Remove parenthesis around (7 + 3w) -8w + 8w2 = 7 + 3w + 2w Combine like terms: 3w + 2w = 5w -8w + 8w2 = 7 + 5w Solving -8w + 8w2 = 7 + 5w Solving for variable 'w'. Reorder the terms: -7 + -8w + -5w + 8w2 = 7 + 5w + -7 + -5w Combine like terms: -8w + -5w = -13w -7 + -13w + 8w2 = 7 + 5w + -7 + -5w Reorder the terms: -7 + -13w + 8w2 = 7 + -7 + 5w + -5w Combine like terms: 7 + -7 = 0 -7 + -13w + 8w2 = 0 + 5w + -5w -7 + -13w + 8w2 = 5w + -5w Combine like terms: 5w + -5w = 0 -7 + -13w + 8w2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.875 + -1.625w + w2 = 0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + -1.625w + 0.875 + w2 = 0 + 0.875 Reorder the terms: -0.875 + 0.875 + -1.625w + w2 = 0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + -1.625w + w2 = 0 + 0.875 -1.625w + w2 = 0 + 0.875 Combine like terms: 0 + 0.875 = 0.875 -1.625w + w2 = 0.875 The w term is -1.625w. Take half its coefficient (-0.8125). Square it (0.66015625) and add it to both sides. Add '0.66015625' to each side of the equation. -1.625w + 0.66015625 + w2 = 0.875 + 0.66015625 Reorder the terms: 0.66015625 + -1.625w + w2 = 0.875 + 0.66015625 Combine like terms: 0.875 + 0.66015625 = 1.53515625 0.66015625 + -1.625w + w2 = 1.53515625 Factor a perfect square on the left side: (w + -0.8125)(w + -0.8125) = 1.53515625 Calculate the square root of the right side: 1.239014225 Break this problem into two subproblems by setting (w + -0.8125) equal to 1.239014225 and -1.239014225.Subproblem 1
w + -0.8125 = 1.239014225 Simplifying w + -0.8125 = 1.239014225 Reorder the terms: -0.8125 + w = 1.239014225 Solving -0.8125 + w = 1.239014225 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.8125' to each side of the equation. -0.8125 + 0.8125 + w = 1.239014225 + 0.8125 Combine like terms: -0.8125 + 0.8125 = 0.0000 0.0000 + w = 1.239014225 + 0.8125 w = 1.239014225 + 0.8125 Combine like terms: 1.239014225 + 0.8125 = 2.051514225 w = 2.051514225 Simplifying w = 2.051514225Subproblem 2
w + -0.8125 = -1.239014225 Simplifying w + -0.8125 = -1.239014225 Reorder the terms: -0.8125 + w = -1.239014225 Solving -0.8125 + w = -1.239014225 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.8125' to each side of the equation. -0.8125 + 0.8125 + w = -1.239014225 + 0.8125 Combine like terms: -0.8125 + 0.8125 = 0.0000 0.0000 + w = -1.239014225 + 0.8125 w = -1.239014225 + 0.8125 Combine like terms: -1.239014225 + 0.8125 = -0.426514225 w = -0.426514225 Simplifying w = -0.426514225Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.051514225, -0.426514225}
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