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Simplifying 4w2 + 3w + -5 = 0 Reorder the terms: -5 + 3w + 4w2 = 0 Solving -5 + 3w + 4w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.25 + 0.75w + w2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + 0.75w + 1.25 + w2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + 0.75w + w2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + 0.75w + w2 = 0 + 1.25 0.75w + w2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 0.75w + w2 = 1.25 The w term is 0.75w. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75w + 0.140625 + w2 = 1.25 + 0.140625 Reorder the terms: 0.140625 + 0.75w + w2 = 1.25 + 0.140625 Combine like terms: 1.25 + 0.140625 = 1.390625 0.140625 + 0.75w + w2 = 1.390625 Factor a perfect square on the left side: (w + 0.375)(w + 0.375) = 1.390625 Calculate the square root of the right side: 1.179247642 Break this problem into two subproblems by setting (w + 0.375) equal to 1.179247642 and -1.179247642.Subproblem 1
w + 0.375 = 1.179247642 Simplifying w + 0.375 = 1.179247642 Reorder the terms: 0.375 + w = 1.179247642 Solving 0.375 + w = 1.179247642 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + w = 1.179247642 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + w = 1.179247642 + -0.375 w = 1.179247642 + -0.375 Combine like terms: 1.179247642 + -0.375 = 0.804247642 w = 0.804247642 Simplifying w = 0.804247642Subproblem 2
w + 0.375 = -1.179247642 Simplifying w + 0.375 = -1.179247642 Reorder the terms: 0.375 + w = -1.179247642 Solving 0.375 + w = -1.179247642 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + w = -1.179247642 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + w = -1.179247642 + -0.375 w = -1.179247642 + -0.375 Combine like terms: -1.179247642 + -0.375 = -1.554247642 w = -1.554247642 Simplifying w = -1.554247642Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.804247642, -1.554247642}
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