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Simplifying g(5 + 2g) = 32 (5 * g + 2g * g) = 32 (5g + 2g2) = 32 Solving 5g + 2g2 = 32 Solving for variable 'g'. Reorder the terms: -32 + 5g + 2g2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + 5g + 2g2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -16 + 2.5g + g2 = 0 Move the constant term to the right: Add '16' to each side of the equation. -16 + 2.5g + 16 + g2 = 0 + 16 Reorder the terms: -16 + 16 + 2.5g + g2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + 2.5g + g2 = 0 + 16 2.5g + g2 = 0 + 16 Combine like terms: 0 + 16 = 16 2.5g + g2 = 16 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 = 16 + 1.5625 Reorder the terms: 1.5625 + 2.5g + g2 = 16 + 1.5625 Combine like terms: 16 + 1.5625 = 17.5625 1.5625 + 2.5g + g2 = 17.5625 Factor a perfect square on the left side: (g + 1.25)(g + 1.25) = 17.5625 Calculate the square root of the right side: 4.190763654 Break this problem into two subproblems by setting (g + 1.25) equal to 4.190763654 and -4.190763654.Subproblem 1
g + 1.25 = 4.190763654 Simplifying g + 1.25 = 4.190763654 Reorder the terms: 1.25 + g = 4.190763654 Solving 1.25 + g = 4.190763654 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 = 4.190763654 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + g = 4.190763654 + -1.25 g = 4.190763654 + -1.25 Combine like terms: 4.190763654 + -1.25 = 2.940763654 g = 2.940763654 Simplifying g = 2.940763654Subproblem 2
g + 1.25 = -4.190763654 Simplifying g + 1.25 = -4.190763654 Reorder the terms: 1.25 + g = -4.190763654 Solving 1.25 + g = -4.190763654 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 = -4.190763654 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + g = -4.190763654 + -1.25 g = -4.190763654 + -1.25 Combine like terms: -4.190763654 + -1.25 = -5.440763654 g = -5.440763654 Simplifying g = -5.440763654Solution
The solution to the problem is based on the solutions from the subproblems. g = {2.940763654, -5.440763654}
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