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Simplifying (-4w2 + 6w + -2) + (7w2 + 3w + 5) = 0 Reorder the terms: (-2 + 6w + -4w2) + (7w2 + 3w + 5) = 0 Remove parenthesis around (-2 + 6w + -4w2) -2 + 6w + -4w2 + (7w2 + 3w + 5) = 0 Reorder the terms: -2 + 6w + -4w2 + (5 + 3w + 7w2) = 0 Remove parenthesis around (5 + 3w + 7w2) -2 + 6w + -4w2 + 5 + 3w + 7w2 = 0 Reorder the terms: -2 + 5 + 6w + 3w + -4w2 + 7w2 = 0 Combine like terms: -2 + 5 = 3 3 + 6w + 3w + -4w2 + 7w2 = 0 Combine like terms: 6w + 3w = 9w 3 + 9w + -4w2 + 7w2 = 0 Combine like terms: -4w2 + 7w2 = 3w2 3 + 9w + 3w2 = 0 Solving 3 + 9w + 3w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '3'. 3(1 + 3w + w2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(1 + 3w + w2)' equal to zero and attempt to solve: Simplifying 1 + 3w + w2 = 0 Solving 1 + 3w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 3w + -1 + w2 = 0 + -1 Reorder the terms: 1 + -1 + 3w + w2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 3w + w2 = 0 + -1 3w + w2 = 0 + -1 Combine like terms: 0 + -1 = -1 3w + w2 = -1 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 + 2.25 Reorder the terms: 2.25 + 3w + w2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + 3w + w2 = 1.25 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (w + 1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
w + 1.5 = 1.118033989 Simplifying w + 1.5 = 1.118033989 Reorder the terms: 1.5 + w = 1.118033989 Solving 1.5 + w = 1.118033989 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 = 1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 1.118033989 + -1.5 w = 1.118033989 + -1.5 Combine like terms: 1.118033989 + -1.5 = -0.381966011 w = -0.381966011 Simplifying w = -0.381966011Subproblem 2
w + 1.5 = -1.118033989 Simplifying w + 1.5 = -1.118033989 Reorder the terms: 1.5 + w = -1.118033989 Solving 1.5 + w = -1.118033989 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 = -1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -1.118033989 + -1.5 w = -1.118033989 + -1.5 Combine like terms: -1.118033989 + -1.5 = -2.618033989 w = -2.618033989 Simplifying w = -2.618033989Solution
The solution to the problem is based on the solutions from the subproblems. w = {-0.381966011, -2.618033989}Solution
w = {-0.381966011, -2.618033989}
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