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Simplifying -12x3 + 32x2 + 23x = 0 Reorder the terms: 23x + 32x2 + -12x3 = 0 Solving 23x + 32x2 + -12x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x'. x(23 + 32x + -12x2) = 0Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(23 + 32x + -12x2)' equal to zero and attempt to solve: Simplifying 23 + 32x + -12x2 = 0 Solving 23 + 32x + -12x2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -1.916666667 + -2.666666667x + x2 = 0 Move the constant term to the right: Add '1.916666667' to each side of the equation. -1.916666667 + -2.666666667x + 1.916666667 + x2 = 0 + 1.916666667 Reorder the terms: -1.916666667 + 1.916666667 + -2.666666667x + x2 = 0 + 1.916666667 Combine like terms: -1.916666667 + 1.916666667 = 0.000000000 0.000000000 + -2.666666667x + x2 = 0 + 1.916666667 -2.666666667x + x2 = 0 + 1.916666667 Combine like terms: 0 + 1.916666667 = 1.916666667 -2.666666667x + x2 = 1.916666667 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 = 1.916666667 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667x + x2 = 1.916666667 + 1.777777780 Combine like terms: 1.916666667 + 1.777777780 = 3.694444447 1.777777780 + -2.666666667x + x2 = 3.694444447 Factor a perfect square on the left side: (x + -1.333333334)(x + -1.333333334) = 3.694444447 Calculate the square root of the right side: 1.922093766 Break this problem into two subproblems by setting (x + -1.333333334) equal to 1.922093766 and -1.922093766.Subproblem 1
x + -1.333333334 = 1.922093766 Simplifying x + -1.333333334 = 1.922093766 Reorder the terms: -1.333333334 + x = 1.922093766 Solving -1.333333334 + x = 1.922093766 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 = 1.922093766 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 1.922093766 + 1.333333334 x = 1.922093766 + 1.333333334 Combine like terms: 1.922093766 + 1.333333334 = 3.2554271 x = 3.2554271 Simplifying x = 3.2554271Subproblem 2
x + -1.333333334 = -1.922093766 Simplifying x + -1.333333334 = -1.922093766 Reorder the terms: -1.333333334 + x = -1.922093766 Solving -1.333333334 + x = -1.922093766 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 = -1.922093766 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -1.922093766 + 1.333333334 x = -1.922093766 + 1.333333334 Combine like terms: -1.922093766 + 1.333333334 = -0.588760432 x = -0.588760432 Simplifying x = -0.588760432Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.2554271, -0.588760432}Solution
x = {0, 3.2554271, -0.588760432}
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