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Simplifying w(2w + -7) = 90 Reorder the terms: w(-7 + 2w) = 90 (-7 * w + 2w * w) = 90 (-7w + 2w2) = 90 Solving -7w + 2w2 = 90 Solving for variable 'w'. Reorder the terms: -90 + -7w + 2w2 = 90 + -90 Combine like terms: 90 + -90 = 0 -90 + -7w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -45 + -3.5w + w2 = 0 Move the constant term to the right: Add '45' to each side of the equation. -45 + -3.5w + 45 + w2 = 0 + 45 Reorder the terms: -45 + 45 + -3.5w + w2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + -3.5w + w2 = 0 + 45 -3.5w + w2 = 0 + 45 Combine like terms: 0 + 45 = 45 -3.5w + w2 = 45 The w term is -3.5w. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5w + 3.0625 + w2 = 45 + 3.0625 Reorder the terms: 3.0625 + -3.5w + w2 = 45 + 3.0625 Combine like terms: 45 + 3.0625 = 48.0625 3.0625 + -3.5w + w2 = 48.0625 Factor a perfect square on the left side: (w + -1.75)(w + -1.75) = 48.0625 Calculate the square root of the right side: 6.932712312 Break this problem into two subproblems by setting (w + -1.75) equal to 6.932712312 and -6.932712312.Subproblem 1
w + -1.75 = 6.932712312 Simplifying w + -1.75 = 6.932712312 Reorder the terms: -1.75 + w = 6.932712312 Solving -1.75 + w = 6.932712312 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + w = 6.932712312 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + w = 6.932712312 + 1.75 w = 6.932712312 + 1.75 Combine like terms: 6.932712312 + 1.75 = 8.682712312 w = 8.682712312 Simplifying w = 8.682712312Subproblem 2
w + -1.75 = -6.932712312 Simplifying w + -1.75 = -6.932712312 Reorder the terms: -1.75 + w = -6.932712312 Solving -1.75 + w = -6.932712312 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + w = -6.932712312 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + w = -6.932712312 + 1.75 w = -6.932712312 + 1.75 Combine like terms: -6.932712312 + 1.75 = -5.182712312 w = -5.182712312 Simplifying w = -5.182712312Solution
The solution to the problem is based on the solutions from the subproblems. w = {8.682712312, -5.182712312}
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