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Simplifying 0 = 2w2 + 5w + -180 Reorder the terms: 0 = -180 + 5w + 2w2 Solving 0 = -180 + 5w + 2w2 Solving for variable 'w'. Combine like terms: 0 + 180 = 180 180 + -5w + -2w2 = -180 + 5w + 2w2 + 180 + -5w + -2w2 Reorder the terms: 180 + -5w + -2w2 = -180 + 180 + 5w + -5w + 2w2 + -2w2 Combine like terms: -180 + 180 = 0 180 + -5w + -2w2 = 0 + 5w + -5w + 2w2 + -2w2 180 + -5w + -2w2 = 5w + -5w + 2w2 + -2w2 Combine like terms: 5w + -5w = 0 180 + -5w + -2w2 = 0 + 2w2 + -2w2 180 + -5w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 180 + -5w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -90 + 2.5w + w2 = 0 Move the constant term to the right: Add '90' to each side of the equation. -90 + 2.5w + 90 + w2 = 0 + 90 Reorder the terms: -90 + 90 + 2.5w + w2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + 2.5w + w2 = 0 + 90 2.5w + w2 = 0 + 90 Combine like terms: 0 + 90 = 90 2.5w + w2 = 90 The w term is 2.5w. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5w + 1.5625 + w2 = 90 + 1.5625 Reorder the terms: 1.5625 + 2.5w + w2 = 90 + 1.5625 Combine like terms: 90 + 1.5625 = 91.5625 1.5625 + 2.5w + w2 = 91.5625 Factor a perfect square on the left side: (w + 1.25)(w + 1.25) = 91.5625 Calculate the square root of the right side: 9.568829605 Break this problem into two subproblems by setting (w + 1.25) equal to 9.568829605 and -9.568829605.Subproblem 1
w + 1.25 = 9.568829605 Simplifying w + 1.25 = 9.568829605 Reorder the terms: 1.25 + w = 9.568829605 Solving 1.25 + w = 9.568829605 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + w = 9.568829605 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + w = 9.568829605 + -1.25 w = 9.568829605 + -1.25 Combine like terms: 9.568829605 + -1.25 = 8.318829605 w = 8.318829605 Simplifying w = 8.318829605Subproblem 2
w + 1.25 = -9.568829605 Simplifying w + 1.25 = -9.568829605 Reorder the terms: 1.25 + w = -9.568829605 Solving 1.25 + w = -9.568829605 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + w = -9.568829605 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + w = -9.568829605 + -1.25 w = -9.568829605 + -1.25 Combine like terms: -9.568829605 + -1.25 = -10.818829605 w = -10.818829605 Simplifying w = -10.818829605Solution
The solution to the problem is based on the solutions from the subproblems. w = {8.318829605, -10.818829605}
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