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Simplifying 15 + 5g(g + 5) = 415 Reorder the terms: 15 + 5g(5 + g) = 415 15 + (5 * 5g + g * 5g) = 415 15 + (25g + 5g2) = 415 Solving 15 + 25g + 5g2 = 415 Solving for variable 'g'. Reorder the terms: 15 + -415 + 25g + 5g2 = 415 + -415 Combine like terms: 15 + -415 = -400 -400 + 25g + 5g2 = 415 + -415 Combine like terms: 415 + -415 = 0 -400 + 25g + 5g2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-80 + 5g + g2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-80 + 5g + g2)' equal to zero and attempt to solve: Simplifying -80 + 5g + g2 = 0 Solving -80 + 5g + g2 = 0 Begin completing the square. Move the constant term to the right: Add '80' to each side of the equation. -80 + 5g + 80 + g2 = 0 + 80 Reorder the terms: -80 + 80 + 5g + g2 = 0 + 80 Combine like terms: -80 + 80 = 0 0 + 5g + g2 = 0 + 80 5g + g2 = 0 + 80 Combine like terms: 0 + 80 = 80 5g + g2 = 80 The g term is 5g. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5g + 6.25 + g2 = 80 + 6.25 Reorder the terms: 6.25 + 5g + g2 = 80 + 6.25 Combine like terms: 80 + 6.25 = 86.25 6.25 + 5g + g2 = 86.25 Factor a perfect square on the left side: (g + 2.5)(g + 2.5) = 86.25 Calculate the square root of the right side: 9.287087811 Break this problem into two subproblems by setting (g + 2.5) equal to 9.287087811 and -9.287087811.Subproblem 1
g + 2.5 = 9.287087811 Simplifying g + 2.5 = 9.287087811 Reorder the terms: 2.5 + g = 9.287087811 Solving 2.5 + g = 9.287087811 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + g = 9.287087811 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + g = 9.287087811 + -2.5 g = 9.287087811 + -2.5 Combine like terms: 9.287087811 + -2.5 = 6.787087811 g = 6.787087811 Simplifying g = 6.787087811Subproblem 2
g + 2.5 = -9.287087811 Simplifying g + 2.5 = -9.287087811 Reorder the terms: 2.5 + g = -9.287087811 Solving 2.5 + g = -9.287087811 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + g = -9.287087811 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + g = -9.287087811 + -2.5 g = -9.287087811 + -2.5 Combine like terms: -9.287087811 + -2.5 = -11.787087811 g = -11.787087811 Simplifying g = -11.787087811Solution
The solution to the problem is based on the solutions from the subproblems. g = {6.787087811, -11.787087811}Solution
g = {6.787087811, -11.787087811}
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