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Simplifying 12x2 + -22x + -114 = 0 Reorder the terms: -114 + -22x + 12x2 = 0 Solving -114 + -22x + 12x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-57 + -11x + 6x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-57 + -11x + 6x2)' equal to zero and attempt to solve: Simplifying -57 + -11x + 6x2 = 0 Solving -57 + -11x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -9.5 + -1.833333333x + x2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + -1.833333333x + 9.5 + x2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + -1.833333333x + x2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + -1.833333333x + x2 = 0 + 9.5 -1.833333333x + x2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 -1.833333333x + x2 = 9.5 The x term is -1.833333333x. Take half its coefficient (-0.9166666665). Square it (0.8402777775) and add it to both sides. Add '0.8402777775' to each side of the equation. -1.833333333x + 0.8402777775 + x2 = 9.5 + 0.8402777775 Reorder the terms: 0.8402777775 + -1.833333333x + x2 = 9.5 + 0.8402777775 Combine like terms: 9.5 + 0.8402777775 = 10.3402777775 0.8402777775 + -1.833333333x + x2 = 10.3402777775 Factor a perfect square on the left side: (x + -0.9166666665)(x + -0.9166666665) = 10.3402777775 Calculate the square root of the right side: 3.21563023 Break this problem into two subproblems by setting (x + -0.9166666665) equal to 3.21563023 and -3.21563023.Subproblem 1
x + -0.9166666665 = 3.21563023 Simplifying x + -0.9166666665 = 3.21563023 Reorder the terms: -0.9166666665 + x = 3.21563023 Solving -0.9166666665 + x = 3.21563023 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + x = 3.21563023 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + x = 3.21563023 + 0.9166666665 x = 3.21563023 + 0.9166666665 Combine like terms: 3.21563023 + 0.9166666665 = 4.1322968965 x = 4.1322968965 Simplifying x = 4.1322968965Subproblem 2
x + -0.9166666665 = -3.21563023 Simplifying x + -0.9166666665 = -3.21563023 Reorder the terms: -0.9166666665 + x = -3.21563023 Solving -0.9166666665 + x = -3.21563023 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + x = -3.21563023 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + x = -3.21563023 + 0.9166666665 x = -3.21563023 + 0.9166666665 Combine like terms: -3.21563023 + 0.9166666665 = -2.2989635635 x = -2.2989635635 Simplifying x = -2.2989635635Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.1322968965, -2.2989635635}Solution
x = {4.1322968965, -2.2989635635}
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