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Simplifying 36g + 6 = 2g(6g + 9) Reorder the terms: 6 + 36g = 2g(6g + 9) Reorder the terms: 6 + 36g = 2g(9 + 6g) 6 + 36g = (9 * 2g + 6g * 2g) 6 + 36g = (18g + 12g2) Solving 6 + 36g = 18g + 12g2 Solving for variable 'g'. Combine like terms: 36g + -18g = 18g 6 + 18g + -12g2 = 18g + 12g2 + -18g + -12g2 Reorder the terms: 6 + 18g + -12g2 = 18g + -18g + 12g2 + -12g2 Combine like terms: 18g + -18g = 0 6 + 18g + -12g2 = 0 + 12g2 + -12g2 6 + 18g + -12g2 = 12g2 + -12g2 Combine like terms: 12g2 + -12g2 = 0 6 + 18g + -12g2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(1 + 3g + -2g2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(1 + 3g + -2g2)' equal to zero and attempt to solve: Simplifying 1 + 3g + -2g2 = 0 Solving 1 + 3g + -2g2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + -1.5g + g2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1.5g + 0.5 + g2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1.5g + g2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1.5g + g2 = 0 + 0.5 -1.5g + g2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1.5g + g2 = 0.5 The g term is -1.5g. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5g + 0.5625 + g2 = 0.5 + 0.5625 Reorder the terms: 0.5625 + -1.5g + g2 = 0.5 + 0.5625 Combine like terms: 0.5 + 0.5625 = 1.0625 0.5625 + -1.5g + g2 = 1.0625 Factor a perfect square on the left side: (g + -0.75)(g + -0.75) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (g + -0.75) equal to 1.030776406 and -1.030776406.Subproblem 1
g + -0.75 = 1.030776406 Simplifying g + -0.75 = 1.030776406 Reorder the terms: -0.75 + g = 1.030776406 Solving -0.75 + g = 1.030776406 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + g = 1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + g = 1.030776406 + 0.75 g = 1.030776406 + 0.75 Combine like terms: 1.030776406 + 0.75 = 1.780776406 g = 1.780776406 Simplifying g = 1.780776406Subproblem 2
g + -0.75 = -1.030776406 Simplifying g + -0.75 = -1.030776406 Reorder the terms: -0.75 + g = -1.030776406 Solving -0.75 + g = -1.030776406 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + g = -1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + g = -1.030776406 + 0.75 g = -1.030776406 + 0.75 Combine like terms: -1.030776406 + 0.75 = -0.280776406 g = -0.280776406 Simplifying g = -0.280776406Solution
The solution to the problem is based on the solutions from the subproblems. g = {1.780776406, -0.280776406}Solution
g = {1.780776406, -0.280776406}
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