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Simplifying 2w(2w + -1) = 21 Reorder the terms: 2w(-1 + 2w) = 21 (-1 * 2w + 2w * 2w) = 21 (-2w + 4w2) = 21 Solving -2w + 4w2 = 21 Solving for variable 'w'. Reorder the terms: -21 + -2w + 4w2 = 21 + -21 Combine like terms: 21 + -21 = 0 -21 + -2w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5.25 + -0.5w + w2 = 0 Move the constant term to the right: Add '5.25' to each side of the equation. -5.25 + -0.5w + 5.25 + w2 = 0 + 5.25 Reorder the terms: -5.25 + 5.25 + -0.5w + w2 = 0 + 5.25 Combine like terms: -5.25 + 5.25 = 0.00 0.00 + -0.5w + w2 = 0 + 5.25 -0.5w + w2 = 0 + 5.25 Combine like terms: 0 + 5.25 = 5.25 -0.5w + w2 = 5.25 The w term is -0.5w. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5w + 0.0625 + w2 = 5.25 + 0.0625 Reorder the terms: 0.0625 + -0.5w + w2 = 5.25 + 0.0625 Combine like terms: 5.25 + 0.0625 = 5.3125 0.0625 + -0.5w + w2 = 5.3125 Factor a perfect square on the left side: (w + -0.25)(w + -0.25) = 5.3125 Calculate the square root of the right side: 2.304886114 Break this problem into two subproblems by setting (w + -0.25) equal to 2.304886114 and -2.304886114.Subproblem 1
w + -0.25 = 2.304886114 Simplifying w + -0.25 = 2.304886114 Reorder the terms: -0.25 + w = 2.304886114 Solving -0.25 + w = 2.304886114 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + w = 2.304886114 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + w = 2.304886114 + 0.25 w = 2.304886114 + 0.25 Combine like terms: 2.304886114 + 0.25 = 2.554886114 w = 2.554886114 Simplifying w = 2.554886114Subproblem 2
w + -0.25 = -2.304886114 Simplifying w + -0.25 = -2.304886114 Reorder the terms: -0.25 + w = -2.304886114 Solving -0.25 + w = -2.304886114 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + w = -2.304886114 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + w = -2.304886114 + 0.25 w = -2.304886114 + 0.25 Combine like terms: -2.304886114 + 0.25 = -2.054886114 w = -2.054886114 Simplifying w = -2.054886114Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.554886114, -2.054886114}
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