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Simplifying 2w2 + -5w + -115 = 0 Reorder the terms: -115 + -5w + 2w2 = 0 Solving -115 + -5w + 2w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -57.5 + -2.5w + w2 = 0 Move the constant term to the right: Add '57.5' to each side of the equation. -57.5 + -2.5w + 57.5 + w2 = 0 + 57.5 Reorder the terms: -57.5 + 57.5 + -2.5w + w2 = 0 + 57.5 Combine like terms: -57.5 + 57.5 = 0.0 0.0 + -2.5w + w2 = 0 + 57.5 -2.5w + w2 = 0 + 57.5 Combine like terms: 0 + 57.5 = 57.5 -2.5w + w2 = 57.5 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 = 57.5 + 1.5625 Reorder the terms: 1.5625 + -2.5w + w2 = 57.5 + 1.5625 Combine like terms: 57.5 + 1.5625 = 59.0625 1.5625 + -2.5w + w2 = 59.0625 Factor a perfect square on the left side: (w + -1.25)(w + -1.25) = 59.0625 Calculate the square root of the right side: 7.685213074 Break this problem into two subproblems by setting (w + -1.25) equal to 7.685213074 and -7.685213074.Subproblem 1
w + -1.25 = 7.685213074 Simplifying w + -1.25 = 7.685213074 Reorder the terms: -1.25 + w = 7.685213074 Solving -1.25 + w = 7.685213074 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 = 7.685213074 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + w = 7.685213074 + 1.25 w = 7.685213074 + 1.25 Combine like terms: 7.685213074 + 1.25 = 8.935213074 w = 8.935213074 Simplifying w = 8.935213074Subproblem 2
w + -1.25 = -7.685213074 Simplifying w + -1.25 = -7.685213074 Reorder the terms: -1.25 + w = -7.685213074 Solving -1.25 + w = -7.685213074 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 = -7.685213074 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + w = -7.685213074 + 1.25 w = -7.685213074 + 1.25 Combine like terms: -7.685213074 + 1.25 = -6.435213074 w = -6.435213074 Simplifying w = -6.435213074Solution
The solution to the problem is based on the solutions from the subproblems. w = {8.935213074, -6.435213074}
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