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Simplifying 8g2 + -9g + 1 = 12g2 + g + -15 Reorder the terms: 1 + -9g + 8g2 = 12g2 + g + -15 Reorder the terms: 1 + -9g + 8g2 = -15 + g + 12g2 Solving 1 + -9g + 8g2 = -15 + g + 12g2 Solving for variable 'g'. Reorder the terms: 1 + 15 + -9g + -1g + 8g2 + -12g2 = -15 + g + 12g2 + 15 + -1g + -12g2 Combine like terms: 1 + 15 = 16 16 + -9g + -1g + 8g2 + -12g2 = -15 + g + 12g2 + 15 + -1g + -12g2 Combine like terms: -9g + -1g = -10g 16 + -10g + 8g2 + -12g2 = -15 + g + 12g2 + 15 + -1g + -12g2 Combine like terms: 8g2 + -12g2 = -4g2 16 + -10g + -4g2 = -15 + g + 12g2 + 15 + -1g + -12g2 Reorder the terms: 16 + -10g + -4g2 = -15 + 15 + g + -1g + 12g2 + -12g2 Combine like terms: -15 + 15 = 0 16 + -10g + -4g2 = 0 + g + -1g + 12g2 + -12g2 16 + -10g + -4g2 = g + -1g + 12g2 + -12g2 Combine like terms: g + -1g = 0 16 + -10g + -4g2 = 0 + 12g2 + -12g2 16 + -10g + -4g2 = 12g2 + -12g2 Combine like terms: 12g2 + -12g2 = 0 16 + -10g + -4g2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(8 + -5g + -2g2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(8 + -5g + -2g2)' equal to zero and attempt to solve: Simplifying 8 + -5g + -2g2 = 0 Solving 8 + -5g + -2g2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -4 + 2.5g + g2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 2.5g + 4 + g2 = 0 + 4 Reorder the terms: -4 + 4 + 2.5g + g2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 2.5g + g2 = 0 + 4 2.5g + g2 = 0 + 4 Combine like terms: 0 + 4 = 4 2.5g + g2 = 4 The g term is 2.5g. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5g + 1.5625 + g2 = 4 + 1.5625 Reorder the terms: 1.5625 + 2.5g + g2 = 4 + 1.5625 Combine like terms: 4 + 1.5625 = 5.5625 1.5625 + 2.5g + g2 = 5.5625 Factor a perfect square on the left side: (g + 1.25)(g + 1.25) = 5.5625 Calculate the square root of the right side: 2.358495283 Break this problem into two subproblems by setting (g + 1.25) equal to 2.358495283 and -2.358495283.Subproblem 1
g + 1.25 = 2.358495283 Simplifying g + 1.25 = 2.358495283 Reorder the terms: 1.25 + g = 2.358495283 Solving 1.25 + g = 2.358495283 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + g = 2.358495283 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + g = 2.358495283 + -1.25 g = 2.358495283 + -1.25 Combine like terms: 2.358495283 + -1.25 = 1.108495283 g = 1.108495283 Simplifying g = 1.108495283Subproblem 2
g + 1.25 = -2.358495283 Simplifying g + 1.25 = -2.358495283 Reorder the terms: 1.25 + g = -2.358495283 Solving 1.25 + g = -2.358495283 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + g = -2.358495283 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + g = -2.358495283 + -1.25 g = -2.358495283 + -1.25 Combine like terms: -2.358495283 + -1.25 = -3.608495283 g = -3.608495283 Simplifying g = -3.608495283Solution
The solution to the problem is based on the solutions from the subproblems. g = {1.108495283, -3.608495283}Solution
g = {1.108495283, -3.608495283}
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