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Simplifying 12x2 + 22x + -18 = 0 Reorder the terms: -18 + 22x + 12x2 = 0 Solving -18 + 22x + 12x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-9 + 11x + 6x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-9 + 11x + 6x2)' equal to zero and attempt to solve: Simplifying -9 + 11x + 6x2 = 0 Solving -9 + 11x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.5 + 1.833333333x + x2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 1.833333333x + 1.5 + x2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 1.833333333x + x2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 1.833333333x + x2 = 0 + 1.5 1.833333333x + x2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 1.833333333x + x2 = 1.5 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 = 1.5 + 0.8402777775 Reorder the terms: 0.8402777775 + 1.833333333x + x2 = 1.5 + 0.8402777775 Combine like terms: 1.5 + 0.8402777775 = 2.3402777775 0.8402777775 + 1.833333333x + x2 = 2.3402777775 Factor a perfect square on the left side: (x + 0.9166666665)(x + 0.9166666665) = 2.3402777775 Calculate the square root of the right side: 1.529796646 Break this problem into two subproblems by setting (x + 0.9166666665) equal to 1.529796646 and -1.529796646.Subproblem 1
x + 0.9166666665 = 1.529796646 Simplifying x + 0.9166666665 = 1.529796646 Reorder the terms: 0.9166666665 + x = 1.529796646 Solving 0.9166666665 + x = 1.529796646 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 = 1.529796646 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = 1.529796646 + -0.9166666665 x = 1.529796646 + -0.9166666665 Combine like terms: 1.529796646 + -0.9166666665 = 0.6131299795 x = 0.6131299795 Simplifying x = 0.6131299795Subproblem 2
x + 0.9166666665 = -1.529796646 Simplifying x + 0.9166666665 = -1.529796646 Reorder the terms: 0.9166666665 + x = -1.529796646 Solving 0.9166666665 + x = -1.529796646 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 = -1.529796646 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = -1.529796646 + -0.9166666665 x = -1.529796646 + -0.9166666665 Combine like terms: -1.529796646 + -0.9166666665 = -2.4464633125 x = -2.4464633125 Simplifying x = -2.4464633125Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.6131299795, -2.4464633125}Solution
x = {0.6131299795, -2.4464633125}
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