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Simplifying 3w + -4w(w + -2) + 1 = 0 Reorder the terms: 3w + -4w(-2 + w) + 1 = 0 3w + (-2 * -4w + w * -4w) + 1 = 0 3w + (8w + -4w2) + 1 = 0 Reorder the terms: 1 + 3w + 8w + -4w2 = 0 Combine like terms: 3w + 8w = 11w 1 + 11w + -4w2 = 0 Solving 1 + 11w + -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'. -0.25 + -2.75w + w2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -2.75w + 0.25 + w2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -2.75w + w2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -2.75w + w2 = 0 + 0.25 -2.75w + w2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -2.75w + w2 = 0.25 The w term is -2.75w. Take half its coefficient (-1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. -2.75w + 1.890625 + w2 = 0.25 + 1.890625 Reorder the terms: 1.890625 + -2.75w + w2 = 0.25 + 1.890625 Combine like terms: 0.25 + 1.890625 = 2.140625 1.890625 + -2.75w + w2 = 2.140625 Factor a perfect square on the left side: (w + -1.375)(w + -1.375) = 2.140625 Calculate the square root of the right side: 1.463087489 Break this problem into two subproblems by setting (w + -1.375) equal to 1.463087489 and -1.463087489.Subproblem 1
w + -1.375 = 1.463087489 Simplifying w + -1.375 = 1.463087489 Reorder the terms: -1.375 + w = 1.463087489 Solving -1.375 + w = 1.463087489 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + w = 1.463087489 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + w = 1.463087489 + 1.375 w = 1.463087489 + 1.375 Combine like terms: 1.463087489 + 1.375 = 2.838087489 w = 2.838087489 Simplifying w = 2.838087489Subproblem 2
w + -1.375 = -1.463087489 Simplifying w + -1.375 = -1.463087489 Reorder the terms: -1.375 + w = -1.463087489 Solving -1.375 + w = -1.463087489 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + w = -1.463087489 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + w = -1.463087489 + 1.375 w = -1.463087489 + 1.375 Combine like terms: -1.463087489 + 1.375 = -0.088087489 w = -0.088087489 Simplifying w = -0.088087489Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.838087489, -0.088087489}
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