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Simplifying (-4w2 + 7w + -4) + (-9w2 + 2w + -2) + -1(-5w2 + 3w + -7) = 0 Reorder the terms: (-4 + 7w + -4w2) + (-9w2 + 2w + -2) + -1(-5w2 + 3w + -7) = 0 Remove parenthesis around (-4 + 7w + -4w2) -4 + 7w + -4w2 + (-9w2 + 2w + -2) + -1(-5w2 + 3w + -7) = 0 Reorder the terms: -4 + 7w + -4w2 + (-2 + 2w + -9w2) + -1(-5w2 + 3w + -7) = 0 Remove parenthesis around (-2 + 2w + -9w2) -4 + 7w + -4w2 + -2 + 2w + -9w2 + -1(-5w2 + 3w + -7) = 0 Reorder the terms: -4 + 7w + -4w2 + -2 + 2w + -9w2 + -1(-7 + 3w + -5w2) = 0 -4 + 7w + -4w2 + -2 + 2w + -9w2 + (-7 * -1 + 3w * -1 + -5w2 * -1) = 0 -4 + 7w + -4w2 + -2 + 2w + -9w2 + (7 + -3w + 5w2) = 0 Reorder the terms: -4 + -2 + 7 + 7w + 2w + -3w + -4w2 + -9w2 + 5w2 = 0 Combine like terms: -4 + -2 = -6 -6 + 7 + 7w + 2w + -3w + -4w2 + -9w2 + 5w2 = 0 Combine like terms: -6 + 7 = 1 1 + 7w + 2w + -3w + -4w2 + -9w2 + 5w2 = 0 Combine like terms: 7w + 2w = 9w 1 + 9w + -3w + -4w2 + -9w2 + 5w2 = 0 Combine like terms: 9w + -3w = 6w 1 + 6w + -4w2 + -9w2 + 5w2 = 0 Combine like terms: -4w2 + -9w2 = -13w2 1 + 6w + -13w2 + 5w2 = 0 Combine like terms: -13w2 + 5w2 = -8w2 1 + 6w + -8w2 = 0 Solving 1 + 6w + -8w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.125 + -0.75w + w2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + -0.75w + 0.125 + w2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + -0.75w + w2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + -0.75w + w2 = 0 + 0.125 -0.75w + w2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 -0.75w + w2 = 0.125 The w term is -0.75w. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75w + 0.140625 + w2 = 0.125 + 0.140625 Reorder the terms: 0.140625 + -0.75w + w2 = 0.125 + 0.140625 Combine like terms: 0.125 + 0.140625 = 0.265625 0.140625 + -0.75w + w2 = 0.265625 Factor a perfect square on the left side: (w + -0.375)(w + -0.375) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (w + -0.375) equal to 0.515388203 and -0.515388203.Subproblem 1
w + -0.375 = 0.515388203 Simplifying w + -0.375 = 0.515388203 Reorder the terms: -0.375 + w = 0.515388203 Solving -0.375 + w = 0.515388203 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + w = 0.515388203 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + w = 0.515388203 + 0.375 w = 0.515388203 + 0.375 Combine like terms: 0.515388203 + 0.375 = 0.890388203 w = 0.890388203 Simplifying w = 0.890388203Subproblem 2
w + -0.375 = -0.515388203 Simplifying w + -0.375 = -0.515388203 Reorder the terms: -0.375 + w = -0.515388203 Solving -0.375 + w = -0.515388203 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + w = -0.515388203 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + w = -0.515388203 + 0.375 w = -0.515388203 + 0.375 Combine like terms: -0.515388203 + 0.375 = -0.140388203 w = -0.140388203 Simplifying w = -0.140388203Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.890388203, -0.140388203}
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