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Simplifying 4g2 + 12g + 7 = 0 Reorder the terms: 7 + 12g + 4g2 = 0 Solving 7 + 12g + 4g2 = 0 Solving for variable 'g'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.75 + 3g + g2 = 0 Move the constant term to the right: Add '-1.75' to each side of the equation. 1.75 + 3g + -1.75 + g2 = 0 + -1.75 Reorder the terms: 1.75 + -1.75 + 3g + g2 = 0 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + 3g + g2 = 0 + -1.75 3g + g2 = 0 + -1.75 Combine like terms: 0 + -1.75 = -1.75 3g + g2 = -1.75 The g term is 3g. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3g + 2.25 + g2 = -1.75 + 2.25 Reorder the terms: 2.25 + 3g + g2 = -1.75 + 2.25 Combine like terms: -1.75 + 2.25 = 0.5 2.25 + 3g + g2 = 0.5 Factor a perfect square on the left side: (g + 1.5)(g + 1.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (g + 1.5) equal to 0.707106781 and -0.707106781.Subproblem 1
g + 1.5 = 0.707106781 Simplifying g + 1.5 = 0.707106781 Reorder the terms: 1.5 + g = 0.707106781 Solving 1.5 + g = 0.707106781 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + g = 0.707106781 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + g = 0.707106781 + -1.5 g = 0.707106781 + -1.5 Combine like terms: 0.707106781 + -1.5 = -0.792893219 g = -0.792893219 Simplifying g = -0.792893219Subproblem 2
g + 1.5 = -0.707106781 Simplifying g + 1.5 = -0.707106781 Reorder the terms: 1.5 + g = -0.707106781 Solving 1.5 + g = -0.707106781 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + g = -0.707106781 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + g = -0.707106781 + -1.5 g = -0.707106781 + -1.5 Combine like terms: -0.707106781 + -1.5 = -2.207106781 g = -2.207106781 Simplifying g = -2.207106781Solution
The solution to the problem is based on the solutions from the subproblems. g = {-0.792893219, -2.207106781}
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