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Simplifying (2w + -3) * w = 45 Reorder the terms: (-3 + 2w) * w = 45 Reorder the terms for easier multiplication: w(-3 + 2w) = 45 (-3 * w + 2w * w) = 45 (-3w + 2w2) = 45 Solving -3w + 2w2 = 45 Solving for variable 'w'. Reorder the terms: -45 + -3w + 2w2 = 45 + -45 Combine like terms: 45 + -45 = 0 -45 + -3w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -22.5 + -1.5w + w2 = 0 Move the constant term to the right: Add '22.5' to each side of the equation. -22.5 + -1.5w + 22.5 + w2 = 0 + 22.5 Reorder the terms: -22.5 + 22.5 + -1.5w + w2 = 0 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + -1.5w + w2 = 0 + 22.5 -1.5w + w2 = 0 + 22.5 Combine like terms: 0 + 22.5 = 22.5 -1.5w + w2 = 22.5 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 = 22.5 + 0.5625 Reorder the terms: 0.5625 + -1.5w + w2 = 22.5 + 0.5625 Combine like terms: 22.5 + 0.5625 = 23.0625 0.5625 + -1.5w + w2 = 23.0625 Factor a perfect square on the left side: (w + -0.75)(w + -0.75) = 23.0625 Calculate the square root of the right side: 4.802343178 Break this problem into two subproblems by setting (w + -0.75) equal to 4.802343178 and -4.802343178.Subproblem 1
w + -0.75 = 4.802343178 Simplifying w + -0.75 = 4.802343178 Reorder the terms: -0.75 + w = 4.802343178 Solving -0.75 + w = 4.802343178 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 = 4.802343178 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + w = 4.802343178 + 0.75 w = 4.802343178 + 0.75 Combine like terms: 4.802343178 + 0.75 = 5.552343178 w = 5.552343178 Simplifying w = 5.552343178Subproblem 2
w + -0.75 = -4.802343178 Simplifying w + -0.75 = -4.802343178 Reorder the terms: -0.75 + w = -4.802343178 Solving -0.75 + w = -4.802343178 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 = -4.802343178 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + w = -4.802343178 + 0.75 w = -4.802343178 + 0.75 Combine like terms: -4.802343178 + 0.75 = -4.052343178 w = -4.052343178 Simplifying w = -4.052343178Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.552343178, -4.052343178}
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