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Simplifying (g + 2)(g + -3) = 1 Reorder the terms: (2 + g)(g + -3) = 1 Reorder the terms: (2 + g)(-3 + g) = 1 Multiply (2 + g) * (-3 + g) (2(-3 + g) + g(-3 + g)) = 1 ((-3 * 2 + g * 2) + g(-3 + g)) = 1 ((-6 + 2g) + g(-3 + g)) = 1 (-6 + 2g + (-3 * g + g * g)) = 1 (-6 + 2g + (-3g + g2)) = 1 Combine like terms: 2g + -3g = -1g (-6 + -1g + g2) = 1 Solving -6 + -1g + g2 = 1 Solving for variable 'g'. Reorder the terms: -6 + -1 + -1g + g2 = 1 + -1 Combine like terms: -6 + -1 = -7 -7 + -1g + g2 = 1 + -1 Combine like terms: 1 + -1 = 0 -7 + -1g + g2 = 0 Begin completing the square. Move the constant term to the right: Add '7' to each side of the equation. -7 + -1g + 7 + g2 = 0 + 7 Reorder the terms: -7 + 7 + -1g + g2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + -1g + g2 = 0 + 7 -1g + g2 = 0 + 7 Combine like terms: 0 + 7 = 7 -1g + g2 = 7 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 = 7 + 0.25 Reorder the terms: 0.25 + -1g + g2 = 7 + 0.25 Combine like terms: 7 + 0.25 = 7.25 0.25 + -1g + g2 = 7.25 Factor a perfect square on the left side: (g + -0.5)(g + -0.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (g + -0.5) equal to 2.692582404 and -2.692582404.Subproblem 1
g + -0.5 = 2.692582404 Simplifying g + -0.5 = 2.692582404 Reorder the terms: -0.5 + g = 2.692582404 Solving -0.5 + g = 2.692582404 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 = 2.692582404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + g = 2.692582404 + 0.5 g = 2.692582404 + 0.5 Combine like terms: 2.692582404 + 0.5 = 3.192582404 g = 3.192582404 Simplifying g = 3.192582404Subproblem 2
g + -0.5 = -2.692582404 Simplifying g + -0.5 = -2.692582404 Reorder the terms: -0.5 + g = -2.692582404 Solving -0.5 + g = -2.692582404 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 = -2.692582404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + g = -2.692582404 + 0.5 g = -2.692582404 + 0.5 Combine like terms: -2.692582404 + 0.5 = -2.192582404 g = -2.192582404 Simplifying g = -2.192582404Solution
The solution to the problem is based on the solutions from the subproblems. g = {3.192582404, -2.192582404}
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