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Simplifying (-2g2 + g + 10) + -1(5g2 + -8g + -3) = 0 Reorder the terms: (10 + g + -2g2) + -1(5g2 + -8g + -3) = 0 Remove parenthesis around (10 + g + -2g2) 10 + g + -2g2 + -1(5g2 + -8g + -3) = 0 Reorder the terms: 10 + g + -2g2 + -1(-3 + -8g + 5g2) = 0 10 + g + -2g2 + (-3 * -1 + -8g * -1 + 5g2 * -1) = 0 10 + g + -2g2 + (3 + 8g + -5g2) = 0 Reorder the terms: 10 + 3 + g + 8g + -2g2 + -5g2 = 0 Combine like terms: 10 + 3 = 13 13 + g + 8g + -2g2 + -5g2 = 0 Combine like terms: g + 8g = 9g 13 + 9g + -2g2 + -5g2 = 0 Combine like terms: -2g2 + -5g2 = -7g2 13 + 9g + -7g2 = 0 Solving 13 + 9g + -7g2 = 0 Solving for variable 'g'. Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -1.857142857 + -1.285714286g + g2 = 0 Move the constant term to the right: Add '1.857142857' to each side of the equation. -1.857142857 + -1.285714286g + 1.857142857 + g2 = 0 + 1.857142857 Reorder the terms: -1.857142857 + 1.857142857 + -1.285714286g + g2 = 0 + 1.857142857 Combine like terms: -1.857142857 + 1.857142857 = 0.000000000 0.000000000 + -1.285714286g + g2 = 0 + 1.857142857 -1.285714286g + g2 = 0 + 1.857142857 Combine like terms: 0 + 1.857142857 = 1.857142857 -1.285714286g + g2 = 1.857142857 The g term is -1.285714286g. 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. -1.285714286g + 0.4132653063 + g2 = 1.857142857 + 0.4132653063 Reorder the terms: 0.4132653063 + -1.285714286g + g2 = 1.857142857 + 0.4132653063 Combine like terms: 1.857142857 + 0.4132653063 = 2.2704081633 0.4132653063 + -1.285714286g + g2 = 2.2704081633 Factor a perfect square on the left side: (g + -0.642857143)(g + -0.642857143) = 2.2704081633 Calculate the square root of the right side: 1.506787365 Break this problem into two subproblems by setting (g + -0.642857143) equal to 1.506787365 and -1.506787365.Subproblem 1
g + -0.642857143 = 1.506787365 Simplifying g + -0.642857143 = 1.506787365 Reorder the terms: -0.642857143 + g = 1.506787365 Solving -0.642857143 + g = 1.506787365 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '0.642857143' to each side of the equation. -0.642857143 + 0.642857143 + g = 1.506787365 + 0.642857143 Combine like terms: -0.642857143 + 0.642857143 = 0.000000000 0.000000000 + g = 1.506787365 + 0.642857143 g = 1.506787365 + 0.642857143 Combine like terms: 1.506787365 + 0.642857143 = 2.149644508 g = 2.149644508 Simplifying g = 2.149644508Subproblem 2
g + -0.642857143 = -1.506787365 Simplifying g + -0.642857143 = -1.506787365 Reorder the terms: -0.642857143 + g = -1.506787365 Solving -0.642857143 + g = -1.506787365 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '0.642857143' to each side of the equation. -0.642857143 + 0.642857143 + g = -1.506787365 + 0.642857143 Combine like terms: -0.642857143 + 0.642857143 = 0.000000000 0.000000000 + g = -1.506787365 + 0.642857143 g = -1.506787365 + 0.642857143 Combine like terms: -1.506787365 + 0.642857143 = -0.863930222 g = -0.863930222 Simplifying g = -0.863930222Solution
The solution to the problem is based on the solutions from the subproblems. g = {2.149644508, -0.863930222}
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