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Simplifying 0 = 15 + -32x + 12x2 Solving 0 = 15 + -32x + 12x2 Solving for variable 'x'. Combine like terms: 0 + -15 = -15 -15 + 32x + -12x2 = 15 + -32x + 12x2 + -15 + 32x + -12x2 Reorder the terms: -15 + 32x + -12x2 = 15 + -15 + -32x + 32x + 12x2 + -12x2 Combine like terms: 15 + -15 = 0 -15 + 32x + -12x2 = 0 + -32x + 32x + 12x2 + -12x2 -15 + 32x + -12x2 = -32x + 32x + 12x2 + -12x2 Combine like terms: -32x + 32x = 0 -15 + 32x + -12x2 = 0 + 12x2 + -12x2 -15 + 32x + -12x2 = 12x2 + -12x2 Combine like terms: 12x2 + -12x2 = 0 -15 + 32x + -12x2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. 1.25 + -2.666666667x + x2 = 0 Move the constant term to the right: Add '-1.25' to each side of the equation. 1.25 + -2.666666667x + -1.25 + x2 = 0 + -1.25 Reorder the terms: 1.25 + -1.25 + -2.666666667x + x2 = 0 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + -2.666666667x + x2 = 0 + -1.25 -2.666666667x + x2 = 0 + -1.25 Combine like terms: 0 + -1.25 = -1.25 -2.666666667x + x2 = -1.25 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 = -1.25 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667x + x2 = -1.25 + 1.777777780 Combine like terms: -1.25 + 1.777777780 = 0.52777778 1.777777780 + -2.666666667x + x2 = 0.52777778 Factor a perfect square on the left side: (x + -1.333333334)(x + -1.333333334) = 0.52777778 Calculate the square root of the right side: 0.726483159 Break this problem into two subproblems by setting (x + -1.333333334) equal to 0.726483159 and -0.726483159.Subproblem 1
x + -1.333333334 = 0.726483159 Simplifying x + -1.333333334 = 0.726483159 Reorder the terms: -1.333333334 + x = 0.726483159 Solving -1.333333334 + x = 0.726483159 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 = 0.726483159 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 0.726483159 + 1.333333334 x = 0.726483159 + 1.333333334 Combine like terms: 0.726483159 + 1.333333334 = 2.059816493 x = 2.059816493 Simplifying x = 2.059816493Subproblem 2
x + -1.333333334 = -0.726483159 Simplifying x + -1.333333334 = -0.726483159 Reorder the terms: -1.333333334 + x = -0.726483159 Solving -1.333333334 + x = -0.726483159 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 = -0.726483159 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -0.726483159 + 1.333333334 x = -0.726483159 + 1.333333334 Combine like terms: -0.726483159 + 1.333333334 = 0.606850175 x = 0.606850175 Simplifying x = 0.606850175Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.059816493, 0.606850175}
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