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Simplifying 12g2 + 14g + -40 = 90 Reorder the terms: -40 + 14g + 12g2 = 90 Solving -40 + 14g + 12g2 = 90 Solving for variable 'g'. Reorder the terms: -40 + -90 + 14g + 12g2 = 90 + -90 Combine like terms: -40 + -90 = -130 -130 + 14g + 12g2 = 90 + -90 Combine like terms: 90 + -90 = 0 -130 + 14g + 12g2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-65 + 7g + 6g2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-65 + 7g + 6g2)' equal to zero and attempt to solve: Simplifying -65 + 7g + 6g2 = 0 Solving -65 + 7g + 6g2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -10.83333333 + 1.166666667g + g2 = 0 Move the constant term to the right: Add '10.83333333' to each side of the equation. -10.83333333 + 1.166666667g + 10.83333333 + g2 = 0 + 10.83333333 Reorder the terms: -10.83333333 + 10.83333333 + 1.166666667g + g2 = 0 + 10.83333333 Combine like terms: -10.83333333 + 10.83333333 = 0.00000000 0.00000000 + 1.166666667g + g2 = 0 + 10.83333333 1.166666667g + g2 = 0 + 10.83333333 Combine like terms: 0 + 10.83333333 = 10.83333333 1.166666667g + g2 = 10.83333333 The g term is 1.166666667g. Take half its coefficient (0.5833333335). Square it (0.3402777780) and add it to both sides. Add '0.3402777780' to each side of the equation. 1.166666667g + 0.3402777780 + g2 = 10.83333333 + 0.3402777780 Reorder the terms: 0.3402777780 + 1.166666667g + g2 = 10.83333333 + 0.3402777780 Combine like terms: 10.83333333 + 0.3402777780 = 11.173611108 0.3402777780 + 1.166666667g + g2 = 11.173611108 Factor a perfect square on the left side: (g + 0.5833333335)(g + 0.5833333335) = 11.173611108 Calculate the square root of the right side: 3.342695186 Break this problem into two subproblems by setting (g + 0.5833333335) equal to 3.342695186 and -3.342695186.Subproblem 1
g + 0.5833333335 = 3.342695186 Simplifying g + 0.5833333335 = 3.342695186 Reorder the terms: 0.5833333335 + g = 3.342695186 Solving 0.5833333335 + g = 3.342695186 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + g = 3.342695186 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + g = 3.342695186 + -0.5833333335 g = 3.342695186 + -0.5833333335 Combine like terms: 3.342695186 + -0.5833333335 = 2.7593618525 g = 2.7593618525 Simplifying g = 2.7593618525Subproblem 2
g + 0.5833333335 = -3.342695186 Simplifying g + 0.5833333335 = -3.342695186 Reorder the terms: 0.5833333335 + g = -3.342695186 Solving 0.5833333335 + g = -3.342695186 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + g = -3.342695186 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + g = -3.342695186 + -0.5833333335 g = -3.342695186 + -0.5833333335 Combine like terms: -3.342695186 + -0.5833333335 = -3.9260285195 g = -3.9260285195 Simplifying g = -3.9260285195Solution
The solution to the problem is based on the solutions from the subproblems. g = {2.7593618525, -3.9260285195}Solution
g = {2.7593618525, -3.9260285195}
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