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Simplifying 12x2 + 11x + -22 = 0 Reorder the terms: -22 + 11x + 12x2 = 0 Solving -22 + 11x + 12x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -1.833333333 + 0.9166666667x + x2 = 0 Move the constant term to the right: Add '1.833333333' to each side of the equation. -1.833333333 + 0.9166666667x + 1.833333333 + x2 = 0 + 1.833333333 Reorder the terms: -1.833333333 + 1.833333333 + 0.9166666667x + x2 = 0 + 1.833333333 Combine like terms: -1.833333333 + 1.833333333 = 0.000000000 0.000000000 + 0.9166666667x + x2 = 0 + 1.833333333 0.9166666667x + x2 = 0 + 1.833333333 Combine like terms: 0 + 1.833333333 = 1.833333333 0.9166666667x + x2 = 1.833333333 The x term is 0.9166666667x. Take half its coefficient (0.4583333334). Square it (0.2100694445) and add it to both sides. Add '0.2100694445' to each side of the equation. 0.9166666667x + 0.2100694445 + x2 = 1.833333333 + 0.2100694445 Reorder the terms: 0.2100694445 + 0.9166666667x + x2 = 1.833333333 + 0.2100694445 Combine like terms: 1.833333333 + 0.2100694445 = 2.0434027775 0.2100694445 + 0.9166666667x + x2 = 2.0434027775 Factor a perfect square on the left side: (x + 0.4583333334)(x + 0.4583333334) = 2.0434027775 Calculate the square root of the right side: 1.4294764 Break this problem into two subproblems by setting (x + 0.4583333334) equal to 1.4294764 and -1.4294764.Subproblem 1
x + 0.4583333334 = 1.4294764 Simplifying x + 0.4583333334 = 1.4294764 Reorder the terms: 0.4583333334 + x = 1.4294764 Solving 0.4583333334 + x = 1.4294764 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + x = 1.4294764 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + x = 1.4294764 + -0.4583333334 x = 1.4294764 + -0.4583333334 Combine like terms: 1.4294764 + -0.4583333334 = 0.9711430666 x = 0.9711430666 Simplifying x = 0.9711430666Subproblem 2
x + 0.4583333334 = -1.4294764 Simplifying x + 0.4583333334 = -1.4294764 Reorder the terms: 0.4583333334 + x = -1.4294764 Solving 0.4583333334 + x = -1.4294764 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + x = -1.4294764 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + x = -1.4294764 + -0.4583333334 x = -1.4294764 + -0.4583333334 Combine like terms: -1.4294764 + -0.4583333334 = -1.8878097334 x = -1.8878097334 Simplifying x = -1.8878097334Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.9711430666, -1.8878097334}
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