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Simplifying 12x3 + -5x2 + -40x = 0 Reorder the terms: -40x + -5x2 + 12x3 = 0 Solving -40x + -5x2 + 12x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x'. x(-40 + -5x + 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 '(-40 + -5x + 12x2)' equal to zero and attempt to solve: Simplifying -40 + -5x + 12x2 = 0 Solving -40 + -5x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -3.333333333 + -0.4166666667x + x2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + -0.4166666667x + 3.333333333 + x2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + -0.4166666667x + x2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + -0.4166666667x + x2 = 0 + 3.333333333 -0.4166666667x + x2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 -0.4166666667x + x2 = 3.333333333 The x term is -0.4166666667x. Take half its coefficient (-0.2083333334). Square it (0.04340277781) and add it to both sides. Add '0.04340277781' to each side of the equation. -0.4166666667x + 0.04340277781 + x2 = 3.333333333 + 0.04340277781 Reorder the terms: 0.04340277781 + -0.4166666667x + x2 = 3.333333333 + 0.04340277781 Combine like terms: 3.333333333 + 0.04340277781 = 3.37673611081 0.04340277781 + -0.4166666667x + x2 = 3.37673611081 Factor a perfect square on the left side: (x + -0.2083333334)(x + -0.2083333334) = 3.37673611081 Calculate the square root of the right side: 1.837589756 Break this problem into two subproblems by setting (x + -0.2083333334) equal to 1.837589756 and -1.837589756.Subproblem 1
x + -0.2083333334 = 1.837589756 Simplifying x + -0.2083333334 = 1.837589756 Reorder the terms: -0.2083333334 + x = 1.837589756 Solving -0.2083333334 + x = 1.837589756 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2083333334' to each side of the equation. -0.2083333334 + 0.2083333334 + x = 1.837589756 + 0.2083333334 Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000 0.0000000000 + x = 1.837589756 + 0.2083333334 x = 1.837589756 + 0.2083333334 Combine like terms: 1.837589756 + 0.2083333334 = 2.0459230894 x = 2.0459230894 Simplifying x = 2.0459230894Subproblem 2
x + -0.2083333334 = -1.837589756 Simplifying x + -0.2083333334 = -1.837589756 Reorder the terms: -0.2083333334 + x = -1.837589756 Solving -0.2083333334 + x = -1.837589756 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2083333334' to each side of the equation. -0.2083333334 + 0.2083333334 + x = -1.837589756 + 0.2083333334 Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000 0.0000000000 + x = -1.837589756 + 0.2083333334 x = -1.837589756 + 0.2083333334 Combine like terms: -1.837589756 + 0.2083333334 = -1.6292564226 x = -1.6292564226 Simplifying x = -1.6292564226Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.0459230894, -1.6292564226}Solution
x = {0, 2.0459230894, -1.6292564226}
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