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Simplifying (3w2 + -5w + -6) + -1(6w2 + 4w + -2) = 0 Reorder the terms: (-6 + -5w + 3w2) + -1(6w2 + 4w + -2) = 0 Remove parenthesis around (-6 + -5w + 3w2) -6 + -5w + 3w2 + -1(6w2 + 4w + -2) = 0 Reorder the terms: -6 + -5w + 3w2 + -1(-2 + 4w + 6w2) = 0 -6 + -5w + 3w2 + (-2 * -1 + 4w * -1 + 6w2 * -1) = 0 -6 + -5w + 3w2 + (2 + -4w + -6w2) = 0 Reorder the terms: -6 + 2 + -5w + -4w + 3w2 + -6w2 = 0 Combine like terms: -6 + 2 = -4 -4 + -5w + -4w + 3w2 + -6w2 = 0 Combine like terms: -5w + -4w = -9w -4 + -9w + 3w2 + -6w2 = 0 Combine like terms: 3w2 + -6w2 = -3w2 -4 + -9w + -3w2 = 0 Solving -4 + -9w + -3w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '-1'. -1(4 + 9w + 3w2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(4 + 9w + 3w2)' equal to zero and attempt to solve: Simplifying 4 + 9w + 3w2 = 0 Solving 4 + 9w + 3w2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + 3w + w2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + 3w + -1.333333333 + w2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + 3w + w2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + 3w + w2 = 0 + -1.333333333 3w + w2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 3w + w2 = -1.333333333 The w term is 3w. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3w + 2.25 + w2 = -1.333333333 + 2.25 Reorder the terms: 2.25 + 3w + w2 = -1.333333333 + 2.25 Combine like terms: -1.333333333 + 2.25 = 0.916666667 2.25 + 3w + w2 = 0.916666667 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 0.916666667 Calculate the square root of the right side: 0.957427108 Break this problem into two subproblems by setting (w + 1.5) equal to 0.957427108 and -0.957427108.Subproblem 1
w + 1.5 = 0.957427108 Simplifying w + 1.5 = 0.957427108 Reorder the terms: 1.5 + w = 0.957427108 Solving 1.5 + w = 0.957427108 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 0.957427108 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 0.957427108 + -1.5 w = 0.957427108 + -1.5 Combine like terms: 0.957427108 + -1.5 = -0.542572892 w = -0.542572892 Simplifying w = -0.542572892Subproblem 2
w + 1.5 = -0.957427108 Simplifying w + 1.5 = -0.957427108 Reorder the terms: 1.5 + w = -0.957427108 Solving 1.5 + w = -0.957427108 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -0.957427108 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -0.957427108 + -1.5 w = -0.957427108 + -1.5 Combine like terms: -0.957427108 + -1.5 = -2.457427108 w = -2.457427108 Simplifying w = -2.457427108Solution
The solution to the problem is based on the solutions from the subproblems. w = {-0.542572892, -2.457427108}Solution
w = {-0.542572892, -2.457427108}
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