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Simplifying (2w + -3) * w = 42 Reorder the terms: (-3 + 2w) * w = 42 Reorder the terms for easier multiplication: w(-3 + 2w) = 42 (-3 * w + 2w * w) = 42 (-3w + 2w2) = 42 Solving -3w + 2w2 = 42 Solving for variable 'w'. Reorder the terms: -42 + -3w + 2w2 = 42 + -42 Combine like terms: 42 + -42 = 0 -42 + -3w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -21 + -1.5w + w2 = 0 Move the constant term to the right: Add '21' to each side of the equation. -21 + -1.5w + 21 + w2 = 0 + 21 Reorder the terms: -21 + 21 + -1.5w + w2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + -1.5w + w2 = 0 + 21 -1.5w + w2 = 0 + 21 Combine like terms: 0 + 21 = 21 -1.5w + w2 = 21 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 = 21 + 0.5625 Reorder the terms: 0.5625 + -1.5w + w2 = 21 + 0.5625 Combine like terms: 21 + 0.5625 = 21.5625 0.5625 + -1.5w + w2 = 21.5625 Factor a perfect square on the left side: (w + -0.75)(w + -0.75) = 21.5625 Calculate the square root of the right side: 4.643543905 Break this problem into two subproblems by setting (w + -0.75) equal to 4.643543905 and -4.643543905.Subproblem 1
w + -0.75 = 4.643543905 Simplifying w + -0.75 = 4.643543905 Reorder the terms: -0.75 + w = 4.643543905 Solving -0.75 + w = 4.643543905 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 = 4.643543905 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + w = 4.643543905 + 0.75 w = 4.643543905 + 0.75 Combine like terms: 4.643543905 + 0.75 = 5.393543905 w = 5.393543905 Simplifying w = 5.393543905Subproblem 2
w + -0.75 = -4.643543905 Simplifying w + -0.75 = -4.643543905 Reorder the terms: -0.75 + w = -4.643543905 Solving -0.75 + w = -4.643543905 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 = -4.643543905 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + w = -4.643543905 + 0.75 w = -4.643543905 + 0.75 Combine like terms: -4.643543905 + 0.75 = -3.893543905 w = -3.893543905 Simplifying w = -3.893543905Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.393543905, -3.893543905}
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