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Simplifying 36g + -90 = 14g2 + 26g + -220 Reorder the terms: -90 + 36g = 14g2 + 26g + -220 Reorder the terms: -90 + 36g = -220 + 26g + 14g2 Solving -90 + 36g = -220 + 26g + 14g2 Solving for variable 'g'. Reorder the terms: -90 + 220 + 36g + -26g + -14g2 = -220 + 26g + 14g2 + 220 + -26g + -14g2 Combine like terms: -90 + 220 = 130 130 + 36g + -26g + -14g2 = -220 + 26g + 14g2 + 220 + -26g + -14g2 Combine like terms: 36g + -26g = 10g 130 + 10g + -14g2 = -220 + 26g + 14g2 + 220 + -26g + -14g2 Reorder the terms: 130 + 10g + -14g2 = -220 + 220 + 26g + -26g + 14g2 + -14g2 Combine like terms: -220 + 220 = 0 130 + 10g + -14g2 = 0 + 26g + -26g + 14g2 + -14g2 130 + 10g + -14g2 = 26g + -26g + 14g2 + -14g2 Combine like terms: 26g + -26g = 0 130 + 10g + -14g2 = 0 + 14g2 + -14g2 130 + 10g + -14g2 = 14g2 + -14g2 Combine like terms: 14g2 + -14g2 = 0 130 + 10g + -14g2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(65 + 5g + -7g2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(65 + 5g + -7g2)' equal to zero and attempt to solve: Simplifying 65 + 5g + -7g2 = 0 Solving 65 + 5g + -7g2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -9.285714286 + -0.7142857143g + g2 = 0 Move the constant term to the right: Add '9.285714286' to each side of the equation. -9.285714286 + -0.7142857143g + 9.285714286 + g2 = 0 + 9.285714286 Reorder the terms: -9.285714286 + 9.285714286 + -0.7142857143g + g2 = 0 + 9.285714286 Combine like terms: -9.285714286 + 9.285714286 = 0.000000000 0.000000000 + -0.7142857143g + g2 = 0 + 9.285714286 -0.7142857143g + g2 = 0 + 9.285714286 Combine like terms: 0 + 9.285714286 = 9.285714286 -0.7142857143g + g2 = 9.285714286 The g term is -0.7142857143g. Take half its coefficient (-0.3571428572). Square it (0.1275510204) and add it to both sides. Add '0.1275510204' to each side of the equation. -0.7142857143g + 0.1275510204 + g2 = 9.285714286 + 0.1275510204 Reorder the terms: 0.1275510204 + -0.7142857143g + g2 = 9.285714286 + 0.1275510204 Combine like terms: 9.285714286 + 0.1275510204 = 9.4132653064 0.1275510204 + -0.7142857143g + g2 = 9.4132653064 Factor a perfect square on the left side: (g + -0.3571428572)(g + -0.3571428572) = 9.4132653064 Calculate the square root of the right side: 3.068104514 Break this problem into two subproblems by setting (g + -0.3571428572) equal to 3.068104514 and -3.068104514.Subproblem 1
g + -0.3571428572 = 3.068104514 Simplifying g + -0.3571428572 = 3.068104514 Reorder the terms: -0.3571428572 + g = 3.068104514 Solving -0.3571428572 + g = 3.068104514 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + g = 3.068104514 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + g = 3.068104514 + 0.3571428572 g = 3.068104514 + 0.3571428572 Combine like terms: 3.068104514 + 0.3571428572 = 3.4252473712 g = 3.4252473712 Simplifying g = 3.4252473712Subproblem 2
g + -0.3571428572 = -3.068104514 Simplifying g + -0.3571428572 = -3.068104514 Reorder the terms: -0.3571428572 + g = -3.068104514 Solving -0.3571428572 + g = -3.068104514 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + g = -3.068104514 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + g = -3.068104514 + 0.3571428572 g = -3.068104514 + 0.3571428572 Combine like terms: -3.068104514 + 0.3571428572 = -2.7109616568 g = -2.7109616568 Simplifying g = -2.7109616568Solution
The solution to the problem is based on the solutions from the subproblems. g = {3.4252473712, -2.7109616568}Solution
g = {3.4252473712, -2.7109616568}
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