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Simplifying (5w2 + 4w + -2) + -1(9w2 + -6w + -5) = 0 Reorder the terms: (-2 + 4w + 5w2) + -1(9w2 + -6w + -5) = 0 Remove parenthesis around (-2 + 4w + 5w2) -2 + 4w + 5w2 + -1(9w2 + -6w + -5) = 0 Reorder the terms: -2 + 4w + 5w2 + -1(-5 + -6w + 9w2) = 0 -2 + 4w + 5w2 + (-5 * -1 + -6w * -1 + 9w2 * -1) = 0 -2 + 4w + 5w2 + (5 + 6w + -9w2) = 0 Reorder the terms: -2 + 5 + 4w + 6w + 5w2 + -9w2 = 0 Combine like terms: -2 + 5 = 3 3 + 4w + 6w + 5w2 + -9w2 = 0 Combine like terms: 4w + 6w = 10w 3 + 10w + 5w2 + -9w2 = 0 Combine like terms: 5w2 + -9w2 = -4w2 3 + 10w + -4w2 = 0 Solving 3 + 10w + -4w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.75 + -2.5w + w2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -2.5w + 0.75 + w2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -2.5w + w2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -2.5w + w2 = 0 + 0.75 -2.5w + w2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -2.5w + w2 = 0.75 The w term is -2.5w. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5w + 1.5625 + w2 = 0.75 + 1.5625 Reorder the terms: 1.5625 + -2.5w + w2 = 0.75 + 1.5625 Combine like terms: 0.75 + 1.5625 = 2.3125 1.5625 + -2.5w + w2 = 2.3125 Factor a perfect square on the left side: (w + -1.25)(w + -1.25) = 2.3125 Calculate the square root of the right side: 1.520690633 Break this problem into two subproblems by setting (w + -1.25) equal to 1.520690633 and -1.520690633.Subproblem 1
w + -1.25 = 1.520690633 Simplifying w + -1.25 = 1.520690633 Reorder the terms: -1.25 + w = 1.520690633 Solving -1.25 + w = 1.520690633 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + w = 1.520690633 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + w = 1.520690633 + 1.25 w = 1.520690633 + 1.25 Combine like terms: 1.520690633 + 1.25 = 2.770690633 w = 2.770690633 Simplifying w = 2.770690633Subproblem 2
w + -1.25 = -1.520690633 Simplifying w + -1.25 = -1.520690633 Reorder the terms: -1.25 + w = -1.520690633 Solving -1.25 + w = -1.520690633 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + w = -1.520690633 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + w = -1.520690633 + 1.25 w = -1.520690633 + 1.25 Combine like terms: -1.520690633 + 1.25 = -0.270690633 w = -0.270690633 Simplifying w = -0.270690633Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.770690633, -0.270690633}
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