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Simplifying (4w2 + -7w + -6) + -1(8w2 + 2w + -3) = 0 Reorder the terms: (-6 + -7w + 4w2) + -1(8w2 + 2w + -3) = 0 Remove parenthesis around (-6 + -7w + 4w2) -6 + -7w + 4w2 + -1(8w2 + 2w + -3) = 0 Reorder the terms: -6 + -7w + 4w2 + -1(-3 + 2w + 8w2) = 0 -6 + -7w + 4w2 + (-3 * -1 + 2w * -1 + 8w2 * -1) = 0 -6 + -7w + 4w2 + (3 + -2w + -8w2) = 0 Reorder the terms: -6 + 3 + -7w + -2w + 4w2 + -8w2 = 0 Combine like terms: -6 + 3 = -3 -3 + -7w + -2w + 4w2 + -8w2 = 0 Combine like terms: -7w + -2w = -9w -3 + -9w + 4w2 + -8w2 = 0 Combine like terms: 4w2 + -8w2 = -4w2 -3 + -9w + -4w2 = 0 Solving -3 + -9w + -4w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '-1'. -1(3 + 9w + 4w2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(3 + 9w + 4w2)' equal to zero and attempt to solve: Simplifying 3 + 9w + 4w2 = 0 Solving 3 + 9w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + 2.25w + w2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + 2.25w + -0.75 + w2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + 2.25w + w2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + 2.25w + w2 = 0 + -0.75 2.25w + w2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 2.25w + w2 = -0.75 The w term is 2.25w. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25w + 1.265625 + w2 = -0.75 + 1.265625 Reorder the terms: 1.265625 + 2.25w + w2 = -0.75 + 1.265625 Combine like terms: -0.75 + 1.265625 = 0.515625 1.265625 + 2.25w + w2 = 0.515625 Factor a perfect square on the left side: (w + 1.125)(w + 1.125) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (w + 1.125) equal to 0.718070331 and -0.718070331.Subproblem 1
w + 1.125 = 0.718070331 Simplifying w + 1.125 = 0.718070331 Reorder the terms: 1.125 + w = 0.718070331 Solving 1.125 + w = 0.718070331 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + w = 0.718070331 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + w = 0.718070331 + -1.125 w = 0.718070331 + -1.125 Combine like terms: 0.718070331 + -1.125 = -0.406929669 w = -0.406929669 Simplifying w = -0.406929669Subproblem 2
w + 1.125 = -0.718070331 Simplifying w + 1.125 = -0.718070331 Reorder the terms: 1.125 + w = -0.718070331 Solving 1.125 + w = -0.718070331 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + w = -0.718070331 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + w = -0.718070331 + -1.125 w = -0.718070331 + -1.125 Combine like terms: -0.718070331 + -1.125 = -1.843070331 w = -1.843070331 Simplifying w = -1.843070331Solution
The solution to the problem is based on the solutions from the subproblems. w = {-0.406929669, -1.843070331}Solution
w = {-0.406929669, -1.843070331}
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