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Simplifying 3 + 7(w + 4) * w = 0 Reorder the terms: 3 + 7(4 + w) * w = 0 Reorder the terms for easier multiplication: 3 + 7w(4 + w) = 0 3 + (4 * 7w + w * 7w) = 0 3 + (28w + 7w2) = 0 Solving 3 + 28w + 7w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 0.4285714286 + 4w + w2 = 0 Move the constant term to the right: Add '-0.4285714286' to each side of the equation. 0.4285714286 + 4w + -0.4285714286 + w2 = 0 + -0.4285714286 Reorder the terms: 0.4285714286 + -0.4285714286 + 4w + w2 = 0 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + 4w + w2 = 0 + -0.4285714286 4w + w2 = 0 + -0.4285714286 Combine like terms: 0 + -0.4285714286 = -0.4285714286 4w + w2 = -0.4285714286 The w term is 4w. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4w + 4 + w2 = -0.4285714286 + 4 Reorder the terms: 4 + 4w + w2 = -0.4285714286 + 4 Combine like terms: -0.4285714286 + 4 = 3.5714285714 4 + 4w + w2 = 3.5714285714 Factor a perfect square on the left side: (w + 2)(w + 2) = 3.5714285714 Calculate the square root of the right side: 1.889822365 Break this problem into two subproblems by setting (w + 2) equal to 1.889822365 and -1.889822365.Subproblem 1
w + 2 = 1.889822365 Simplifying w + 2 = 1.889822365 Reorder the terms: 2 + w = 1.889822365 Solving 2 + w = 1.889822365 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + w = 1.889822365 + -2 Combine like terms: 2 + -2 = 0 0 + w = 1.889822365 + -2 w = 1.889822365 + -2 Combine like terms: 1.889822365 + -2 = -0.110177635 w = -0.110177635 Simplifying w = -0.110177635Subproblem 2
w + 2 = -1.889822365 Simplifying w + 2 = -1.889822365 Reorder the terms: 2 + w = -1.889822365 Solving 2 + w = -1.889822365 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + w = -1.889822365 + -2 Combine like terms: 2 + -2 = 0 0 + w = -1.889822365 + -2 w = -1.889822365 + -2 Combine like terms: -1.889822365 + -2 = -3.889822365 w = -3.889822365 Simplifying w = -3.889822365Solution
The solution to the problem is based on the solutions from the subproblems. w = {-0.110177635, -3.889822365}
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