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Simplifying 4g + -45g + 7 + -38g(9 + g + -12) = -83g2 + 4 Reorder the terms: 4g + -45g + 7 + -38g(9 + -12 + g) = -83g2 + 4 Combine like terms: 9 + -12 = -3 4g + -45g + 7 + -38g(-3 + g) = -83g2 + 4 4g + -45g + 7 + (-3 * -38g + g * -38g) = -83g2 + 4 4g + -45g + 7 + (114g + -38g2) = -83g2 + 4 Reorder the terms: 7 + 4g + -45g + 114g + -38g2 = -83g2 + 4 Combine like terms: 4g + -45g = -41g 7 + -41g + 114g + -38g2 = -83g2 + 4 Combine like terms: -41g + 114g = 73g 7 + 73g + -38g2 = -83g2 + 4 Reorder the terms: 7 + 73g + -38g2 = 4 + -83g2 Solving 7 + 73g + -38g2 = 4 + -83g2 Solving for variable 'g'. Reorder the terms: 7 + -4 + 73g + -38g2 + 83g2 = 4 + -83g2 + -4 + 83g2 Combine like terms: 7 + -4 = 3 3 + 73g + -38g2 + 83g2 = 4 + -83g2 + -4 + 83g2 Combine like terms: -38g2 + 83g2 = 45g2 3 + 73g + 45g2 = 4 + -83g2 + -4 + 83g2 Reorder the terms: 3 + 73g + 45g2 = 4 + -4 + -83g2 + 83g2 Combine like terms: 4 + -4 = 0 3 + 73g + 45g2 = 0 + -83g2 + 83g2 3 + 73g + 45g2 = -83g2 + 83g2 Combine like terms: -83g2 + 83g2 = 0 3 + 73g + 45g2 = 0 Begin completing the square. Divide all terms by 45 the coefficient of the squared term: Divide each side by '45'. 0.06666666667 + 1.622222222g + g2 = 0 Move the constant term to the right: Add '-0.06666666667' to each side of the equation. 0.06666666667 + 1.622222222g + -0.06666666667 + g2 = 0 + -0.06666666667 Reorder the terms: 0.06666666667 + -0.06666666667 + 1.622222222g + g2 = 0 + -0.06666666667 Combine like terms: 0.06666666667 + -0.06666666667 = 0.00000000000 0.00000000000 + 1.622222222g + g2 = 0 + -0.06666666667 1.622222222g + g2 = 0 + -0.06666666667 Combine like terms: 0 + -0.06666666667 = -0.06666666667 1.622222222g + g2 = -0.06666666667 The g term is 1.622222222g. Take half its coefficient (0.811111111). Square it (0.6579012344) and add it to both sides. Add '0.6579012344' to each side of the equation. 1.622222222g + 0.6579012344 + g2 = -0.06666666667 + 0.6579012344 Reorder the terms: 0.6579012344 + 1.622222222g + g2 = -0.06666666667 + 0.6579012344 Combine like terms: -0.06666666667 + 0.6579012344 = 0.59123456773 0.6579012344 + 1.622222222g + g2 = 0.59123456773 Factor a perfect square on the left side: (g + 0.811111111)(g + 0.811111111) = 0.59123456773 Calculate the square root of the right side: 0.76891779 Break this problem into two subproblems by setting (g + 0.811111111) equal to 0.76891779 and -0.76891779.Subproblem 1
g + 0.811111111 = 0.76891779 Simplifying g + 0.811111111 = 0.76891779 Reorder the terms: 0.811111111 + g = 0.76891779 Solving 0.811111111 + g = 0.76891779 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-0.811111111' to each side of the equation. 0.811111111 + -0.811111111 + g = 0.76891779 + -0.811111111 Combine like terms: 0.811111111 + -0.811111111 = 0.000000000 0.000000000 + g = 0.76891779 + -0.811111111 g = 0.76891779 + -0.811111111 Combine like terms: 0.76891779 + -0.811111111 = -0.042193321 g = -0.042193321 Simplifying g = -0.042193321Subproblem 2
g + 0.811111111 = -0.76891779 Simplifying g + 0.811111111 = -0.76891779 Reorder the terms: 0.811111111 + g = -0.76891779 Solving 0.811111111 + g = -0.76891779 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-0.811111111' to each side of the equation. 0.811111111 + -0.811111111 + g = -0.76891779 + -0.811111111 Combine like terms: 0.811111111 + -0.811111111 = 0.000000000 0.000000000 + g = -0.76891779 + -0.811111111 g = -0.76891779 + -0.811111111 Combine like terms: -0.76891779 + -0.811111111 = -1.580028901 g = -1.580028901 Simplifying g = -1.580028901Solution
The solution to the problem is based on the solutions from the subproblems. g = {-0.042193321, -1.580028901}
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