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Simplifying 12x2 + 32x + -48 = 0 Reorder the terms: -48 + 32x + 12x2 = 0 Solving -48 + 32x + 12x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '4'. 4(-12 + 8x + 3x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-12 + 8x + 3x2)' equal to zero and attempt to solve: Simplifying -12 + 8x + 3x2 = 0 Solving -12 + 8x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -4 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 2.666666667x + 4 + x2 = 0 + 4 Reorder the terms: -4 + 4 + 2.666666667x + x2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 2.666666667x + x2 = 0 + 4 2.666666667x + x2 = 0 + 4 Combine like terms: 0 + 4 = 4 2.666666667x + x2 = 4 The x term is 2.666666667x. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667x + 1.777777780 + x2 = 4 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = 4 + 1.777777780 Combine like terms: 4 + 1.777777780 = 5.77777778 1.777777780 + 2.666666667x + x2 = 5.77777778 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 5.77777778 Calculate the square root of the right side: 2.403700851 Break this problem into two subproblems by setting (x + 1.333333334) equal to 2.403700851 and -2.403700851.Subproblem 1
x + 1.333333334 = 2.403700851 Simplifying x + 1.333333334 = 2.403700851 Reorder the terms: 1.333333334 + x = 2.403700851 Solving 1.333333334 + x = 2.403700851 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 2.403700851 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 2.403700851 + -1.333333334 x = 2.403700851 + -1.333333334 Combine like terms: 2.403700851 + -1.333333334 = 1.070367517 x = 1.070367517 Simplifying x = 1.070367517Subproblem 2
x + 1.333333334 = -2.403700851 Simplifying x + 1.333333334 = -2.403700851 Reorder the terms: 1.333333334 + x = -2.403700851 Solving 1.333333334 + x = -2.403700851 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -2.403700851 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -2.403700851 + -1.333333334 x = -2.403700851 + -1.333333334 Combine like terms: -2.403700851 + -1.333333334 = -3.737034185 x = -3.737034185 Simplifying x = -3.737034185Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.070367517, -3.737034185}Solution
x = {1.070367517, -3.737034185}
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