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Simplifying 12x2 + -68x + 66 = 0 Reorder the terms: 66 + -68x + 12x2 = 0 Solving 66 + -68x + 12x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(33 + -34x + 6x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(33 + -34x + 6x2)' equal to zero and attempt to solve: Simplifying 33 + -34x + 6x2 = 0 Solving 33 + -34x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 5.5 + -5.666666667x + x2 = 0 Move the constant term to the right: Add '-5.5' to each side of the equation. 5.5 + -5.666666667x + -5.5 + x2 = 0 + -5.5 Reorder the terms: 5.5 + -5.5 + -5.666666667x + x2 = 0 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + -5.666666667x + x2 = 0 + -5.5 -5.666666667x + x2 = 0 + -5.5 Combine like terms: 0 + -5.5 = -5.5 -5.666666667x + x2 = -5.5 The x term is -5.666666667x. Take half its coefficient (-2.833333334). Square it (8.027777782) and add it to both sides. Add '8.027777782' to each side of the equation. -5.666666667x + 8.027777782 + x2 = -5.5 + 8.027777782 Reorder the terms: 8.027777782 + -5.666666667x + x2 = -5.5 + 8.027777782 Combine like terms: -5.5 + 8.027777782 = 2.527777782 8.027777782 + -5.666666667x + x2 = 2.527777782 Factor a perfect square on the left side: (x + -2.833333334)(x + -2.833333334) = 2.527777782 Calculate the square root of the right side: 1.58989867 Break this problem into two subproblems by setting (x + -2.833333334) equal to 1.58989867 and -1.58989867.Subproblem 1
x + -2.833333334 = 1.58989867 Simplifying x + -2.833333334 = 1.58989867 Reorder the terms: -2.833333334 + x = 1.58989867 Solving -2.833333334 + x = 1.58989867 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.833333334' to each side of the equation. -2.833333334 + 2.833333334 + x = 1.58989867 + 2.833333334 Combine like terms: -2.833333334 + 2.833333334 = 0.000000000 0.000000000 + x = 1.58989867 + 2.833333334 x = 1.58989867 + 2.833333334 Combine like terms: 1.58989867 + 2.833333334 = 4.423232004 x = 4.423232004 Simplifying x = 4.423232004Subproblem 2
x + -2.833333334 = -1.58989867 Simplifying x + -2.833333334 = -1.58989867 Reorder the terms: -2.833333334 + x = -1.58989867 Solving -2.833333334 + x = -1.58989867 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.833333334' to each side of the equation. -2.833333334 + 2.833333334 + x = -1.58989867 + 2.833333334 Combine like terms: -2.833333334 + 2.833333334 = 0.000000000 0.000000000 + x = -1.58989867 + 2.833333334 x = -1.58989867 + 2.833333334 Combine like terms: -1.58989867 + 2.833333334 = 1.243434664 x = 1.243434664 Simplifying x = 1.243434664Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.423232004, 1.243434664}Solution
x = {4.423232004, 1.243434664}
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