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Simplifying 12x2 + 3x + 5 = 120 Reorder the terms: 5 + 3x + 12x2 = 120 Solving 5 + 3x + 12x2 = 120 Solving for variable 'x'. Reorder the terms: 5 + -120 + 3x + 12x2 = 120 + -120 Combine like terms: 5 + -120 = -115 -115 + 3x + 12x2 = 120 + -120 Combine like terms: 120 + -120 = 0 -115 + 3x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -9.583333333 + 0.25x + x2 = 0 Move the constant term to the right: Add '9.583333333' to each side of the equation. -9.583333333 + 0.25x + 9.583333333 + x2 = 0 + 9.583333333 Reorder the terms: -9.583333333 + 9.583333333 + 0.25x + x2 = 0 + 9.583333333 Combine like terms: -9.583333333 + 9.583333333 = 0.000000000 0.000000000 + 0.25x + x2 = 0 + 9.583333333 0.25x + x2 = 0 + 9.583333333 Combine like terms: 0 + 9.583333333 = 9.583333333 0.25x + x2 = 9.583333333 The x term is 0.25x. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25x + 0.015625 + x2 = 9.583333333 + 0.015625 Reorder the terms: 0.015625 + 0.25x + x2 = 9.583333333 + 0.015625 Combine like terms: 9.583333333 + 0.015625 = 9.598958333 0.015625 + 0.25x + x2 = 9.598958333 Factor a perfect square on the left side: (x + 0.125)(x + 0.125) = 9.598958333 Calculate the square root of the right side: 3.098218574 Break this problem into two subproblems by setting (x + 0.125) equal to 3.098218574 and -3.098218574.Subproblem 1
x + 0.125 = 3.098218574 Simplifying x + 0.125 = 3.098218574 Reorder the terms: 0.125 + x = 3.098218574 Solving 0.125 + x = 3.098218574 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = 3.098218574 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = 3.098218574 + -0.125 x = 3.098218574 + -0.125 Combine like terms: 3.098218574 + -0.125 = 2.973218574 x = 2.973218574 Simplifying x = 2.973218574Subproblem 2
x + 0.125 = -3.098218574 Simplifying x + 0.125 = -3.098218574 Reorder the terms: 0.125 + x = -3.098218574 Solving 0.125 + x = -3.098218574 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = -3.098218574 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = -3.098218574 + -0.125 x = -3.098218574 + -0.125 Combine like terms: -3.098218574 + -0.125 = -3.223218574 x = -3.223218574 Simplifying x = -3.223218574Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.973218574, -3.223218574}
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