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Simplifying w2 + w + -108 = 0 Reorder the terms: -108 + w + w2 = 0 Solving -108 + w + w2 = 0 Solving for variable 'w'. Begin completing the square. Move the constant term to the right: Add '108' to each side of the equation. -108 + w + 108 + w2 = 0 + 108 Reorder the terms: -108 + 108 + w + w2 = 0 + 108 Combine like terms: -108 + 108 = 0 0 + w + w2 = 0 + 108 w + w2 = 0 + 108 Combine like terms: 0 + 108 = 108 w + w2 = 108 The w term is w. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. w + 0.25 + w2 = 108 + 0.25 Reorder the terms: 0.25 + w + w2 = 108 + 0.25 Combine like terms: 108 + 0.25 = 108.25 0.25 + w + w2 = 108.25 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 108.25 Calculate the square root of the right side: 10.404326023 Break this problem into two subproblems by setting (w + 0.5) equal to 10.404326023 and -10.404326023.Subproblem 1
w + 0.5 = 10.404326023 Simplifying w + 0.5 = 10.404326023 Reorder the terms: 0.5 + w = 10.404326023 Solving 0.5 + w = 10.404326023 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 10.404326023 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 10.404326023 + -0.5 w = 10.404326023 + -0.5 Combine like terms: 10.404326023 + -0.5 = 9.904326023 w = 9.904326023 Simplifying w = 9.904326023Subproblem 2
w + 0.5 = -10.404326023 Simplifying w + 0.5 = -10.404326023 Reorder the terms: 0.5 + w = -10.404326023 Solving 0.5 + w = -10.404326023 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -10.404326023 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -10.404326023 + -0.5 w = -10.404326023 + -0.5 Combine like terms: -10.404326023 + -0.5 = -10.904326023 w = -10.904326023 Simplifying w = -10.904326023Solution
The solution to the problem is based on the solutions from the subproblems. w = {9.904326023, -10.904326023}
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