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Simplifying 18x3 + 48x2 + -32x = 0 Reorder the terms: -32x + 48x2 + 18x3 = 0 Solving -32x + 48x2 + 18x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2x'. 2x(-16 + 24x + 9x2) = 0 Ignore the factor 2.Subproblem 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 '(-16 + 24x + 9x2)' equal to zero and attempt to solve: Simplifying -16 + 24x + 9x2 = 0 Solving -16 + 24x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.777777778 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '1.777777778' to each side of the equation. -1.777777778 + 2.666666667x + 1.777777778 + x2 = 0 + 1.777777778 Reorder the terms: -1.777777778 + 1.777777778 + 2.666666667x + x2 = 0 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + 2.666666667x + x2 = 0 + 1.777777778 2.666666667x + x2 = 0 + 1.777777778 Combine like terms: 0 + 1.777777778 = 1.777777778 2.666666667x + x2 = 1.777777778 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 = 1.777777778 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = 1.777777778 + 1.777777780 Combine like terms: 1.777777778 + 1.777777780 = 3.555555558 1.777777780 + 2.666666667x + x2 = 3.555555558 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 3.555555558 Calculate the square root of the right side: 1.885618084 Break this problem into two subproblems by setting (x + 1.333333334) equal to 1.885618084 and -1.885618084.Subproblem 1
x + 1.333333334 = 1.885618084 Simplifying x + 1.333333334 = 1.885618084 Reorder the terms: 1.333333334 + x = 1.885618084 Solving 1.333333334 + x = 1.885618084 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 = 1.885618084 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 1.885618084 + -1.333333334 x = 1.885618084 + -1.333333334 Combine like terms: 1.885618084 + -1.333333334 = 0.55228475 x = 0.55228475 Simplifying x = 0.55228475Subproblem 2
x + 1.333333334 = -1.885618084 Simplifying x + 1.333333334 = -1.885618084 Reorder the terms: 1.333333334 + x = -1.885618084 Solving 1.333333334 + x = -1.885618084 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 = -1.885618084 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -1.885618084 + -1.333333334 x = -1.885618084 + -1.333333334 Combine like terms: -1.885618084 + -1.333333334 = -3.218951418 x = -3.218951418 Simplifying x = -3.218951418Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.55228475, -3.218951418}Solution
x = {0, 0.55228475, -3.218951418}
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