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Simplifying 2(g + 5) = g(g + 5) + g2 Reorder the terms: 2(5 + g) = g(g + 5) + g2 (5 * 2 + g * 2) = g(g + 5) + g2 (10 + 2g) = g(g + 5) + g2 Reorder the terms: 10 + 2g = g(5 + g) + g2 10 + 2g = (5 * g + g * g) + g2 10 + 2g = (5g + g2) + g2 Combine like terms: g2 + g2 = 2g2 10 + 2g = 5g + 2g2 Solving 10 + 2g = 5g + 2g2 Solving for variable 'g'. Combine like terms: 2g + -5g = -3g 10 + -3g + -2g2 = 5g + 2g2 + -5g + -2g2 Reorder the terms: 10 + -3g + -2g2 = 5g + -5g + 2g2 + -2g2 Combine like terms: 5g + -5g = 0 10 + -3g + -2g2 = 0 + 2g2 + -2g2 10 + -3g + -2g2 = 2g2 + -2g2 Combine like terms: 2g2 + -2g2 = 0 10 + -3g + -2g2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -5 + 1.5g + g2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 1.5g + 5 + g2 = 0 + 5 Reorder the terms: -5 + 5 + 1.5g + g2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 1.5g + g2 = 0 + 5 1.5g + g2 = 0 + 5 Combine like terms: 0 + 5 = 5 1.5g + g2 = 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 = 5 + 0.5625 Reorder the terms: 0.5625 + 1.5g + g2 = 5 + 0.5625 Combine like terms: 5 + 0.5625 = 5.5625 0.5625 + 1.5g + g2 = 5.5625 Factor a perfect square on the left side: (g + 0.75)(g + 0.75) = 5.5625 Calculate the square root of the right side: 2.358495283 Break this problem into two subproblems by setting (g + 0.75) equal to 2.358495283 and -2.358495283.Subproblem 1
g + 0.75 = 2.358495283 Simplifying g + 0.75 = 2.358495283 Reorder the terms: 0.75 + g = 2.358495283 Solving 0.75 + g = 2.358495283 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 = 2.358495283 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + g = 2.358495283 + -0.75 g = 2.358495283 + -0.75 Combine like terms: 2.358495283 + -0.75 = 1.608495283 g = 1.608495283 Simplifying g = 1.608495283Subproblem 2
g + 0.75 = -2.358495283 Simplifying g + 0.75 = -2.358495283 Reorder the terms: 0.75 + g = -2.358495283 Solving 0.75 + g = -2.358495283 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 = -2.358495283 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + g = -2.358495283 + -0.75 g = -2.358495283 + -0.75 Combine like terms: -2.358495283 + -0.75 = -3.108495283 g = -3.108495283 Simplifying g = -3.108495283Solution
The solution to the problem is based on the solutions from the subproblems. g = {1.608495283, -3.108495283}
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