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Simplifying (8w2 + -8w + -7) + -1(4w2 + 6w + -5) = 0 Reorder the terms: (-7 + -8w + 8w2) + -1(4w2 + 6w + -5) = 0 Remove parenthesis around (-7 + -8w + 8w2) -7 + -8w + 8w2 + -1(4w2 + 6w + -5) = 0 Reorder the terms: -7 + -8w + 8w2 + -1(-5 + 6w + 4w2) = 0 -7 + -8w + 8w2 + (-5 * -1 + 6w * -1 + 4w2 * -1) = 0 -7 + -8w + 8w2 + (5 + -6w + -4w2) = 0 Reorder the terms: -7 + 5 + -8w + -6w + 8w2 + -4w2 = 0 Combine like terms: -7 + 5 = -2 -2 + -8w + -6w + 8w2 + -4w2 = 0 Combine like terms: -8w + -6w = -14w -2 + -14w + 8w2 + -4w2 = 0 Combine like terms: 8w2 + -4w2 = 4w2 -2 + -14w + 4w2 = 0 Solving -2 + -14w + 4w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + -7w + 2w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + -7w + 2w2)' equal to zero and attempt to solve: Simplifying -1 + -7w + 2w2 = 0 Solving -1 + -7w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + -3.5w + w2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -3.5w + 0.5 + w2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -3.5w + w2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -3.5w + w2 = 0 + 0.5 -3.5w + w2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -3.5w + w2 = 0.5 The w term is -3.5w. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5w + 3.0625 + w2 = 0.5 + 3.0625 Reorder the terms: 3.0625 + -3.5w + w2 = 0.5 + 3.0625 Combine like terms: 0.5 + 3.0625 = 3.5625 3.0625 + -3.5w + w2 = 3.5625 Factor a perfect square on the left side: (w + -1.75)(w + -1.75) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (w + -1.75) equal to 1.887458609 and -1.887458609.Subproblem 1
w + -1.75 = 1.887458609 Simplifying w + -1.75 = 1.887458609 Reorder the terms: -1.75 + w = 1.887458609 Solving -1.75 + w = 1.887458609 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + w = 1.887458609 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + w = 1.887458609 + 1.75 w = 1.887458609 + 1.75 Combine like terms: 1.887458609 + 1.75 = 3.637458609 w = 3.637458609 Simplifying w = 3.637458609Subproblem 2
w + -1.75 = -1.887458609 Simplifying w + -1.75 = -1.887458609 Reorder the terms: -1.75 + w = -1.887458609 Solving -1.75 + w = -1.887458609 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + w = -1.887458609 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + w = -1.887458609 + 1.75 w = -1.887458609 + 1.75 Combine like terms: -1.887458609 + 1.75 = -0.137458609 w = -0.137458609 Simplifying w = -0.137458609Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.637458609, -0.137458609}Solution
w = {3.637458609, -0.137458609}
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