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Simplifying 144 = (2w + 3) * w Reorder the terms: 144 = (3 + 2w) * w Reorder the terms for easier multiplication: 144 = w(3 + 2w) 144 = (3 * w + 2w * w) 144 = (3w + 2w2) Solving 144 = 3w + 2w2 Solving for variable 'w'. Reorder the terms: 144 + -3w + -2w2 = 3w + -3w + 2w2 + -2w2 Combine like terms: 3w + -3w = 0 144 + -3w + -2w2 = 0 + 2w2 + -2w2 144 + -3w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 144 + -3w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -72 + 1.5w + w2 = 0 Move the constant term to the right: Add '72' to each side of the equation. -72 + 1.5w + 72 + w2 = 0 + 72 Reorder the terms: -72 + 72 + 1.5w + w2 = 0 + 72 Combine like terms: -72 + 72 = 0 0 + 1.5w + w2 = 0 + 72 1.5w + w2 = 0 + 72 Combine like terms: 0 + 72 = 72 1.5w + w2 = 72 The w term is 1.5w. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5w + 0.5625 + w2 = 72 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 72 + 0.5625 Combine like terms: 72 + 0.5625 = 72.5625 0.5625 + 1.5w + w2 = 72.5625 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 72.5625 Calculate the square root of the right side: 8.518362519 Break this problem into two subproblems by setting (w + 0.75) equal to 8.518362519 and -8.518362519.Subproblem 1
w + 0.75 = 8.518362519 Simplifying w + 0.75 = 8.518362519 Reorder the terms: 0.75 + w = 8.518362519 Solving 0.75 + w = 8.518362519 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = 8.518362519 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 8.518362519 + -0.75 w = 8.518362519 + -0.75 Combine like terms: 8.518362519 + -0.75 = 7.768362519 w = 7.768362519 Simplifying w = 7.768362519Subproblem 2
w + 0.75 = -8.518362519 Simplifying w + 0.75 = -8.518362519 Reorder the terms: 0.75 + w = -8.518362519 Solving 0.75 + w = -8.518362519 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = -8.518362519 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -8.518362519 + -0.75 w = -8.518362519 + -0.75 Combine like terms: -8.518362519 + -0.75 = -9.268362519 w = -9.268362519 Simplifying w = -9.268362519Solution
The solution to the problem is based on the solutions from the subproblems. w = {7.768362519, -9.268362519}
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