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Simplifying 5g2 + -3g + 6 = 2g2 + 9 Reorder the terms: 6 + -3g + 5g2 = 2g2 + 9 Reorder the terms: 6 + -3g + 5g2 = 9 + 2g2 Solving 6 + -3g + 5g2 = 9 + 2g2 Solving for variable 'g'. Reorder the terms: 6 + -9 + -3g + 5g2 + -2g2 = 9 + 2g2 + -9 + -2g2 Combine like terms: 6 + -9 = -3 -3 + -3g + 5g2 + -2g2 = 9 + 2g2 + -9 + -2g2 Combine like terms: 5g2 + -2g2 = 3g2 -3 + -3g + 3g2 = 9 + 2g2 + -9 + -2g2 Reorder the terms: -3 + -3g + 3g2 = 9 + -9 + 2g2 + -2g2 Combine like terms: 9 + -9 = 0 -3 + -3g + 3g2 = 0 + 2g2 + -2g2 -3 + -3g + 3g2 = 2g2 + -2g2 Combine like terms: 2g2 + -2g2 = 0 -3 + -3g + 3g2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-1 + -1g + g2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-1 + -1g + g2)' equal to zero and attempt to solve: Simplifying -1 + -1g + g2 = 0 Solving -1 + -1g + g2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -1g + 1 + g2 = 0 + 1 Reorder the terms: -1 + 1 + -1g + g2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1g + g2 = 0 + 1 -1g + g2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1g + g2 = 1 The g term is -1g. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1g + 0.25 + g2 = 1 + 0.25 Reorder the terms: 0.25 + -1g + g2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + -1g + g2 = 1.25 Factor a perfect square on the left side: (g + -0.5)(g + -0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (g + -0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
g + -0.5 = 1.118033989 Simplifying g + -0.5 = 1.118033989 Reorder the terms: -0.5 + g = 1.118033989 Solving -0.5 + g = 1.118033989 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + g = 1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + g = 1.118033989 + 0.5 g = 1.118033989 + 0.5 Combine like terms: 1.118033989 + 0.5 = 1.618033989 g = 1.618033989 Simplifying g = 1.618033989Subproblem 2
g + -0.5 = -1.118033989 Simplifying g + -0.5 = -1.118033989 Reorder the terms: -0.5 + g = -1.118033989 Solving -0.5 + g = -1.118033989 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + g = -1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + g = -1.118033989 + 0.5 g = -1.118033989 + 0.5 Combine like terms: -1.118033989 + 0.5 = -0.618033989 g = -0.618033989 Simplifying g = -0.618033989Solution
The solution to the problem is based on the solutions from the subproblems. g = {1.618033989, -0.618033989}Solution
g = {1.618033989, -0.618033989}
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