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Simplifying w(w + -4) + -3w(3w + -2) = (w + -4) Reorder the terms: w(-4 + w) + -3w(3w + -2) = (w + -4) (-4 * w + w * w) + -3w(3w + -2) = (w + -4) (-4w + w2) + -3w(3w + -2) = (w + -4) Reorder the terms: -4w + w2 + -3w(-2 + 3w) = (w + -4) -4w + w2 + (-2 * -3w + 3w * -3w) = (w + -4) -4w + w2 + (6w + -9w2) = (w + -4) Reorder the terms: -4w + 6w + w2 + -9w2 = (w + -4) Combine like terms: -4w + 6w = 2w 2w + w2 + -9w2 = (w + -4) Combine like terms: w2 + -9w2 = -8w2 2w + -8w2 = (w + -4) Reorder the terms: 2w + -8w2 = (-4 + w) Remove parenthesis around (-4 + w) 2w + -8w2 = -4 + w Solving 2w + -8w2 = -4 + w Solving for variable 'w'. Reorder the terms: 4 + 2w + -1w + -8w2 = -4 + w + 4 + -1w Combine like terms: 2w + -1w = 1w 4 + 1w + -8w2 = -4 + w + 4 + -1w Reorder the terms: 4 + 1w + -8w2 = -4 + 4 + w + -1w Combine like terms: -4 + 4 = 0 4 + 1w + -8w2 = 0 + w + -1w 4 + 1w + -8w2 = w + -1w Combine like terms: w + -1w = 0 4 + 1w + -8w2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.5 + -0.125w + w2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -0.125w + 0.5 + w2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -0.125w + w2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -0.125w + w2 = 0 + 0.5 -0.125w + w2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -0.125w + w2 = 0.5 The w term is -0.125w. Take half its coefficient (-0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. -0.125w + 0.00390625 + w2 = 0.5 + 0.00390625 Reorder the terms: 0.00390625 + -0.125w + w2 = 0.5 + 0.00390625 Combine like terms: 0.5 + 0.00390625 = 0.50390625 0.00390625 + -0.125w + w2 = 0.50390625 Factor a perfect square on the left side: (w + -0.0625)(w + -0.0625) = 0.50390625 Calculate the square root of the right side: 0.709863543 Break this problem into two subproblems by setting (w + -0.0625) equal to 0.709863543 and -0.709863543.Subproblem 1
w + -0.0625 = 0.709863543 Simplifying w + -0.0625 = 0.709863543 Reorder the terms: -0.0625 + w = 0.709863543 Solving -0.0625 + w = 0.709863543 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + w = 0.709863543 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + w = 0.709863543 + 0.0625 w = 0.709863543 + 0.0625 Combine like terms: 0.709863543 + 0.0625 = 0.772363543 w = 0.772363543 Simplifying w = 0.772363543Subproblem 2
w + -0.0625 = -0.709863543 Simplifying w + -0.0625 = -0.709863543 Reorder the terms: -0.0625 + w = -0.709863543 Solving -0.0625 + w = -0.709863543 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + w = -0.709863543 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + w = -0.709863543 + 0.0625 w = -0.709863543 + 0.0625 Combine like terms: -0.709863543 + 0.0625 = -0.647363543 w = -0.647363543 Simplifying w = -0.647363543Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.772363543, -0.647363543}
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