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Simplifying 4g2 = 4g + 1 Reorder the terms: 4g2 = 1 + 4g Solving 4g2 = 1 + 4g Solving for variable 'g'. Reorder the terms: -1 + -4g + 4g2 = 1 + 4g + -1 + -4g Reorder the terms: -1 + -4g + 4g2 = 1 + -1 + 4g + -4g Combine like terms: 1 + -1 = 0 -1 + -4g + 4g2 = 0 + 4g + -4g -1 + -4g + 4g2 = 4g + -4g Combine like terms: 4g + -4g = 0 -1 + -4g + 4g2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + -1g + g2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1g + 0.25 + g2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1g + g2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1g + g2 = 0 + 0.25 -1g + g2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1g + g2 = 0.25 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 = 0.25 + 0.25 Reorder the terms: 0.25 + -1g + g2 = 0.25 + 0.25 Combine like terms: 0.25 + 0.25 = 0.5 0.25 + -1g + g2 = 0.5 Factor a perfect square on the left side: (g + -0.5)(g + -0.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (g + -0.5) equal to 0.707106781 and -0.707106781.Subproblem 1
g + -0.5 = 0.707106781 Simplifying g + -0.5 = 0.707106781 Reorder the terms: -0.5 + g = 0.707106781 Solving -0.5 + g = 0.707106781 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 = 0.707106781 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + g = 0.707106781 + 0.5 g = 0.707106781 + 0.5 Combine like terms: 0.707106781 + 0.5 = 1.207106781 g = 1.207106781 Simplifying g = 1.207106781Subproblem 2
g + -0.5 = -0.707106781 Simplifying g + -0.5 = -0.707106781 Reorder the terms: -0.5 + g = -0.707106781 Solving -0.5 + g = -0.707106781 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 = -0.707106781 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + g = -0.707106781 + 0.5 g = -0.707106781 + 0.5 Combine like terms: -0.707106781 + 0.5 = -0.207106781 g = -0.207106781 Simplifying g = -0.207106781Solution
The solution to the problem is based on the solutions from the subproblems. g = {1.207106781, -0.207106781}
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