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Simplifying (6w2 + -5w + 2) + -1(-1w2 + 4w + 3) = 0 Reorder the terms: (2 + -5w + 6w2) + -1(-1w2 + 4w + 3) = 0 Remove parenthesis around (2 + -5w + 6w2) 2 + -5w + 6w2 + -1(-1w2 + 4w + 3) = 0 Reorder the terms: 2 + -5w + 6w2 + -1(3 + 4w + -1w2) = 0 2 + -5w + 6w2 + (3 * -1 + 4w * -1 + -1w2 * -1) = 0 2 + -5w + 6w2 + (-3 + -4w + 1w2) = 0 Reorder the terms: 2 + -3 + -5w + -4w + 6w2 + 1w2 = 0 Combine like terms: 2 + -3 = -1 -1 + -5w + -4w + 6w2 + 1w2 = 0 Combine like terms: -5w + -4w = -9w -1 + -9w + 6w2 + 1w2 = 0 Combine like terms: 6w2 + 1w2 = 7w2 -1 + -9w + 7w2 = 0 Solving -1 + -9w + 7w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.1428571429 + -1.285714286w + w2 = 0 Move the constant term to the right: Add '0.1428571429' to each side of the equation. -0.1428571429 + -1.285714286w + 0.1428571429 + w2 = 0 + 0.1428571429 Reorder the terms: -0.1428571429 + 0.1428571429 + -1.285714286w + w2 = 0 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + -1.285714286w + w2 = 0 + 0.1428571429 -1.285714286w + w2 = 0 + 0.1428571429 Combine like terms: 0 + 0.1428571429 = 0.1428571429 -1.285714286w + w2 = 0.1428571429 The w term is -1.285714286w. Take half its coefficient (-0.642857143). Square it (0.4132653063) and add it to both sides. Add '0.4132653063' to each side of the equation. -1.285714286w + 0.4132653063 + w2 = 0.1428571429 + 0.4132653063 Reorder the terms: 0.4132653063 + -1.285714286w + w2 = 0.1428571429 + 0.4132653063 Combine like terms: 0.1428571429 + 0.4132653063 = 0.5561224492 0.4132653063 + -1.285714286w + w2 = 0.5561224492 Factor a perfect square on the left side: (w + -0.642857143)(w + -0.642857143) = 0.5561224492 Calculate the square root of the right side: 0.745736179 Break this problem into two subproblems by setting (w + -0.642857143) equal to 0.745736179 and -0.745736179.Subproblem 1
w + -0.642857143 = 0.745736179 Simplifying w + -0.642857143 = 0.745736179 Reorder the terms: -0.642857143 + w = 0.745736179 Solving -0.642857143 + w = 0.745736179 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.642857143' to each side of the equation. -0.642857143 + 0.642857143 + w = 0.745736179 + 0.642857143 Combine like terms: -0.642857143 + 0.642857143 = 0.000000000 0.000000000 + w = 0.745736179 + 0.642857143 w = 0.745736179 + 0.642857143 Combine like terms: 0.745736179 + 0.642857143 = 1.388593322 w = 1.388593322 Simplifying w = 1.388593322Subproblem 2
w + -0.642857143 = -0.745736179 Simplifying w + -0.642857143 = -0.745736179 Reorder the terms: -0.642857143 + w = -0.745736179 Solving -0.642857143 + w = -0.745736179 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.642857143' to each side of the equation. -0.642857143 + 0.642857143 + w = -0.745736179 + 0.642857143 Combine like terms: -0.642857143 + 0.642857143 = 0.000000000 0.000000000 + w = -0.745736179 + 0.642857143 w = -0.745736179 + 0.642857143 Combine like terms: -0.745736179 + 0.642857143 = -0.102879036 w = -0.102879036 Simplifying w = -0.102879036Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.388593322, -0.102879036}
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