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Simplifying 5g(3g + 5) + 3(g + -7) = 0 Reorder the terms: 5g(5 + 3g) + 3(g + -7) = 0 (5 * 5g + 3g * 5g) + 3(g + -7) = 0 (25g + 15g2) + 3(g + -7) = 0 Reorder the terms: 25g + 15g2 + 3(-7 + g) = 0 25g + 15g2 + (-7 * 3 + g * 3) = 0 25g + 15g2 + (-21 + 3g) = 0 Reorder the terms: -21 + 25g + 3g + 15g2 = 0 Combine like terms: 25g + 3g = 28g -21 + 28g + 15g2 = 0 Solving -21 + 28g + 15g2 = 0 Solving for variable 'g'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -1.4 + 1.866666667g + g2 = 0 Move the constant term to the right: Add '1.4' to each side of the equation. -1.4 + 1.866666667g + 1.4 + g2 = 0 + 1.4 Reorder the terms: -1.4 + 1.4 + 1.866666667g + g2 = 0 + 1.4 Combine like terms: -1.4 + 1.4 = 0.0 0.0 + 1.866666667g + g2 = 0 + 1.4 1.866666667g + g2 = 0 + 1.4 Combine like terms: 0 + 1.4 = 1.4 1.866666667g + g2 = 1.4 The g term is 1.866666667g. Take half its coefficient (0.9333333335). Square it (0.8711111114) and add it to both sides. Add '0.8711111114' to each side of the equation. 1.866666667g + 0.8711111114 + g2 = 1.4 + 0.8711111114 Reorder the terms: 0.8711111114 + 1.866666667g + g2 = 1.4 + 0.8711111114 Combine like terms: 1.4 + 0.8711111114 = 2.2711111114 0.8711111114 + 1.866666667g + g2 = 2.2711111114 Factor a perfect square on the left side: (g + 0.9333333335)(g + 0.9333333335) = 2.2711111114 Calculate the square root of the right side: 1.507020607 Break this problem into two subproblems by setting (g + 0.9333333335) equal to 1.507020607 and -1.507020607.Subproblem 1
g + 0.9333333335 = 1.507020607 Simplifying g + 0.9333333335 = 1.507020607 Reorder the terms: 0.9333333335 + g = 1.507020607 Solving 0.9333333335 + g = 1.507020607 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-0.9333333335' to each side of the equation. 0.9333333335 + -0.9333333335 + g = 1.507020607 + -0.9333333335 Combine like terms: 0.9333333335 + -0.9333333335 = 0.0000000000 0.0000000000 + g = 1.507020607 + -0.9333333335 g = 1.507020607 + -0.9333333335 Combine like terms: 1.507020607 + -0.9333333335 = 0.5736872735 g = 0.5736872735 Simplifying g = 0.5736872735Subproblem 2
g + 0.9333333335 = -1.507020607 Simplifying g + 0.9333333335 = -1.507020607 Reorder the terms: 0.9333333335 + g = -1.507020607 Solving 0.9333333335 + g = -1.507020607 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-0.9333333335' to each side of the equation. 0.9333333335 + -0.9333333335 + g = -1.507020607 + -0.9333333335 Combine like terms: 0.9333333335 + -0.9333333335 = 0.0000000000 0.0000000000 + g = -1.507020607 + -0.9333333335 g = -1.507020607 + -0.9333333335 Combine like terms: -1.507020607 + -0.9333333335 = -2.4403539405 g = -2.4403539405 Simplifying g = -2.4403539405Solution
The solution to the problem is based on the solutions from the subproblems. g = {0.5736872735, -2.4403539405}
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