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Simplifying (5w2 + -4w + -6) + (7w2 + -2w + 3) = 0 Reorder the terms: (-6 + -4w + 5w2) + (7w2 + -2w + 3) = 0 Remove parenthesis around (-6 + -4w + 5w2) -6 + -4w + 5w2 + (7w2 + -2w + 3) = 0 Reorder the terms: -6 + -4w + 5w2 + (3 + -2w + 7w2) = 0 Remove parenthesis around (3 + -2w + 7w2) -6 + -4w + 5w2 + 3 + -2w + 7w2 = 0 Reorder the terms: -6 + 3 + -4w + -2w + 5w2 + 7w2 = 0 Combine like terms: -6 + 3 = -3 -3 + -4w + -2w + 5w2 + 7w2 = 0 Combine like terms: -4w + -2w = -6w -3 + -6w + 5w2 + 7w2 = 0 Combine like terms: 5w2 + 7w2 = 12w2 -3 + -6w + 12w2 = 0 Solving -3 + -6w + 12w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '3'. 3(-1 + -2w + 4w2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-1 + -2w + 4w2)' equal to zero and attempt to solve: Simplifying -1 + -2w + 4w2 = 0 Solving -1 + -2w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + -0.5w + w2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -0.5w + 0.25 + w2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -0.5w + w2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -0.5w + w2 = 0 + 0.25 -0.5w + w2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -0.5w + w2 = 0.25 The w term is -0.5w. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5w + 0.0625 + w2 = 0.25 + 0.0625 Reorder the terms: 0.0625 + -0.5w + w2 = 0.25 + 0.0625 Combine like terms: 0.25 + 0.0625 = 0.3125 0.0625 + -0.5w + w2 = 0.3125 Factor a perfect square on the left side: (w + -0.25)(w + -0.25) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (w + -0.25) equal to 0.559016994 and -0.559016994.Subproblem 1
w + -0.25 = 0.559016994 Simplifying w + -0.25 = 0.559016994 Reorder the terms: -0.25 + w = 0.559016994 Solving -0.25 + w = 0.559016994 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + w = 0.559016994 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + w = 0.559016994 + 0.25 w = 0.559016994 + 0.25 Combine like terms: 0.559016994 + 0.25 = 0.809016994 w = 0.809016994 Simplifying w = 0.809016994Subproblem 2
w + -0.25 = -0.559016994 Simplifying w + -0.25 = -0.559016994 Reorder the terms: -0.25 + w = -0.559016994 Solving -0.25 + w = -0.559016994 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + w = -0.559016994 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + w = -0.559016994 + 0.25 w = -0.559016994 + 0.25 Combine like terms: -0.559016994 + 0.25 = -0.309016994 w = -0.309016994 Simplifying w = -0.309016994Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.809016994, -0.309016994}Solution
w = {0.809016994, -0.309016994}
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