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Simplifying 110 = w(w + 15) Reorder the terms: 110 = w(15 + w) 110 = (15 * w + w * w) 110 = (15w + w2) Solving 110 = 15w + w2 Solving for variable 'w'. Reorder the terms: 110 + -15w + -1w2 = 15w + -15w + w2 + -1w2 Combine like terms: 15w + -15w = 0 110 + -15w + -1w2 = 0 + w2 + -1w2 110 + -15w + -1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 110 + -15w + -1w2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -110 + 15w + w2 = 0 Move the constant term to the right: Add '110' to each side of the equation. -110 + 15w + 110 + w2 = 0 + 110 Reorder the terms: -110 + 110 + 15w + w2 = 0 + 110 Combine like terms: -110 + 110 = 0 0 + 15w + w2 = 0 + 110 15w + w2 = 0 + 110 Combine like terms: 0 + 110 = 110 15w + w2 = 110 The w term is 15w. Take half its coefficient (7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. 15w + 56.25 + w2 = 110 + 56.25 Reorder the terms: 56.25 + 15w + w2 = 110 + 56.25 Combine like terms: 110 + 56.25 = 166.25 56.25 + 15w + w2 = 166.25 Factor a perfect square on the left side: (w + 7.5)(w + 7.5) = 166.25 Calculate the square root of the right side: 12.893796958 Break this problem into two subproblems by setting (w + 7.5) equal to 12.893796958 and -12.893796958.Subproblem 1
w + 7.5 = 12.893796958 Simplifying w + 7.5 = 12.893796958 Reorder the terms: 7.5 + w = 12.893796958 Solving 7.5 + w = 12.893796958 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + w = 12.893796958 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + w = 12.893796958 + -7.5 w = 12.893796958 + -7.5 Combine like terms: 12.893796958 + -7.5 = 5.393796958 w = 5.393796958 Simplifying w = 5.393796958Subproblem 2
w + 7.5 = -12.893796958 Simplifying w + 7.5 = -12.893796958 Reorder the terms: 7.5 + w = -12.893796958 Solving 7.5 + w = -12.893796958 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + w = -12.893796958 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + w = -12.893796958 + -7.5 w = -12.893796958 + -7.5 Combine like terms: -12.893796958 + -7.5 = -20.393796958 w = -20.393796958 Simplifying w = -20.393796958Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.393796958, -20.393796958}
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