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Simplifying 5(5g + 1) * g = 1 Reorder the terms: 5(1 + 5g) * g = 1 Reorder the terms for easier multiplication: 5g(1 + 5g) = 1 (1 * 5g + 5g * 5g) = 1 (5g + 25g2) = 1 Solving 5g + 25g2 = 1 Solving for variable 'g'. Reorder the terms: -1 + 5g + 25g2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 5g + 25g2 = 0 Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. -0.04 + 0.2g + g2 = 0 Move the constant term to the right: Add '0.04' to each side of the equation. -0.04 + 0.2g + 0.04 + g2 = 0 + 0.04 Reorder the terms: -0.04 + 0.04 + 0.2g + g2 = 0 + 0.04 Combine like terms: -0.04 + 0.04 = 0.00 0.00 + 0.2g + g2 = 0 + 0.04 0.2g + g2 = 0 + 0.04 Combine like terms: 0 + 0.04 = 0.04 0.2g + g2 = 0.04 The g term is 0.2g. Take half its coefficient (0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. 0.2g + 0.01 + g2 = 0.04 + 0.01 Reorder the terms: 0.01 + 0.2g + g2 = 0.04 + 0.01 Combine like terms: 0.04 + 0.01 = 0.05 0.01 + 0.2g + g2 = 0.05 Factor a perfect square on the left side: (g + 0.1)(g + 0.1) = 0.05 Calculate the square root of the right side: 0.223606798 Break this problem into two subproblems by setting (g + 0.1) equal to 0.223606798 and -0.223606798.Subproblem 1
g + 0.1 = 0.223606798 Simplifying g + 0.1 = 0.223606798 Reorder the terms: 0.1 + g = 0.223606798 Solving 0.1 + g = 0.223606798 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + g = 0.223606798 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + g = 0.223606798 + -0.1 g = 0.223606798 + -0.1 Combine like terms: 0.223606798 + -0.1 = 0.123606798 g = 0.123606798 Simplifying g = 0.123606798Subproblem 2
g + 0.1 = -0.223606798 Simplifying g + 0.1 = -0.223606798 Reorder the terms: 0.1 + g = -0.223606798 Solving 0.1 + g = -0.223606798 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + g = -0.223606798 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + g = -0.223606798 + -0.1 g = -0.223606798 + -0.1 Combine like terms: -0.223606798 + -0.1 = -0.323606798 g = -0.323606798 Simplifying g = -0.323606798Solution
The solution to the problem is based on the solutions from the subproblems. g = {0.123606798, -0.323606798}
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