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Simplifying 12x3 + 16x2 + -18x = 0 Reorder the terms: -18x + 16x2 + 12x3 = 0 Solving -18x + 16x2 + 12x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2x'. 2x(-9 + 8x + 6x2) = 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 '(-9 + 8x + 6x2)' equal to zero and attempt to solve: Simplifying -9 + 8x + 6x2 = 0 Solving -9 + 8x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.5 + 1.333333333x + x2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 1.333333333x + 1.5 + x2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 1.333333333x + x2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 1.333333333x + x2 = 0 + 1.5 1.333333333x + x2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 1.333333333x + x2 = 1.5 The x term is 1.333333333x. Take half its coefficient (0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. 1.333333333x + 0.4444444442 + x2 = 1.5 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333x + x2 = 1.5 + 0.4444444442 Combine like terms: 1.5 + 0.4444444442 = 1.9444444442 0.4444444442 + 1.333333333x + x2 = 1.9444444442 Factor a perfect square on the left side: (x + 0.6666666665)(x + 0.6666666665) = 1.9444444442 Calculate the square root of the right side: 1.394433377 Break this problem into two subproblems by setting (x + 0.6666666665) equal to 1.394433377 and -1.394433377.Subproblem 1
x + 0.6666666665 = 1.394433377 Simplifying x + 0.6666666665 = 1.394433377 Reorder the terms: 0.6666666665 + x = 1.394433377 Solving 0.6666666665 + x = 1.394433377 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = 1.394433377 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 1.394433377 + -0.6666666665 x = 1.394433377 + -0.6666666665 Combine like terms: 1.394433377 + -0.6666666665 = 0.7277667105 x = 0.7277667105 Simplifying x = 0.7277667105Subproblem 2
x + 0.6666666665 = -1.394433377 Simplifying x + 0.6666666665 = -1.394433377 Reorder the terms: 0.6666666665 + x = -1.394433377 Solving 0.6666666665 + x = -1.394433377 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = -1.394433377 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -1.394433377 + -0.6666666665 x = -1.394433377 + -0.6666666665 Combine like terms: -1.394433377 + -0.6666666665 = -2.0611000435 x = -2.0611000435 Simplifying x = -2.0611000435Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7277667105, -2.0611000435}Solution
x = {0, 0.7277667105, -2.0611000435}
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