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Simplifying (-2w2 + -4w + 9) + (-3w2 + 5w + -1) + -1(-6w2 + 7w + 7) = 0 Reorder the terms: (9 + -4w + -2w2) + (-3w2 + 5w + -1) + -1(-6w2 + 7w + 7) = 0 Remove parenthesis around (9 + -4w + -2w2) 9 + -4w + -2w2 + (-3w2 + 5w + -1) + -1(-6w2 + 7w + 7) = 0 Reorder the terms: 9 + -4w + -2w2 + (-1 + 5w + -3w2) + -1(-6w2 + 7w + 7) = 0 Remove parenthesis around (-1 + 5w + -3w2) 9 + -4w + -2w2 + -1 + 5w + -3w2 + -1(-6w2 + 7w + 7) = 0 Reorder the terms: 9 + -4w + -2w2 + -1 + 5w + -3w2 + -1(7 + 7w + -6w2) = 0 9 + -4w + -2w2 + -1 + 5w + -3w2 + (7 * -1 + 7w * -1 + -6w2 * -1) = 0 9 + -4w + -2w2 + -1 + 5w + -3w2 + (-7 + -7w + 6w2) = 0 Reorder the terms: 9 + -1 + -7 + -4w + 5w + -7w + -2w2 + -3w2 + 6w2 = 0 Combine like terms: 9 + -1 = 8 8 + -7 + -4w + 5w + -7w + -2w2 + -3w2 + 6w2 = 0 Combine like terms: 8 + -7 = 1 1 + -4w + 5w + -7w + -2w2 + -3w2 + 6w2 = 0 Combine like terms: -4w + 5w = 1w 1 + 1w + -7w + -2w2 + -3w2 + 6w2 = 0 Combine like terms: 1w + -7w = -6w 1 + -6w + -2w2 + -3w2 + 6w2 = 0 Combine like terms: -2w2 + -3w2 = -5w2 1 + -6w + -5w2 + 6w2 = 0 Combine like terms: -5w2 + 6w2 = 1w2 1 + -6w + 1w2 = 0 Solving 1 + -6w + 1w2 = 0 Solving for variable 'w'. Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -6w + -1 + w2 = 0 + -1 Reorder the terms: 1 + -1 + -6w + w2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -6w + w2 = 0 + -1 -6w + w2 = 0 + -1 Combine like terms: 0 + -1 = -1 -6w + w2 = -1 The w term is -6w. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6w + 9 + w2 = -1 + 9 Reorder the terms: 9 + -6w + w2 = -1 + 9 Combine like terms: -1 + 9 = 8 9 + -6w + w2 = 8 Factor a perfect square on the left side: (w + -3)(w + -3) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (w + -3) equal to 2.828427125 and -2.828427125.Subproblem 1
w + -3 = 2.828427125 Simplifying w + -3 = 2.828427125 Reorder the terms: -3 + w = 2.828427125 Solving -3 + w = 2.828427125 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + w = 2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + w = 2.828427125 + 3 w = 2.828427125 + 3 Combine like terms: 2.828427125 + 3 = 5.828427125 w = 5.828427125 Simplifying w = 5.828427125Subproblem 2
w + -3 = -2.828427125 Simplifying w + -3 = -2.828427125 Reorder the terms: -3 + w = -2.828427125 Solving -3 + w = -2.828427125 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + w = -2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + w = -2.828427125 + 3 w = -2.828427125 + 3 Combine like terms: -2.828427125 + 3 = 0.171572875 w = 0.171572875 Simplifying w = 0.171572875Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.828427125, 0.171572875}
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