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Simplifying 12x3 + -54x2 + -40x = 0 Reorder the terms: -40x + -54x2 + 12x3 = 0 Solving -40x + -54x2 + 12x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2x'. 2x(-20 + -27x + 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 '(-20 + -27x + 6x2)' equal to zero and attempt to solve: Simplifying -20 + -27x + 6x2 = 0 Solving -20 + -27x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -3.333333333 + -4.5x + x2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + -4.5x + 3.333333333 + x2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + -4.5x + x2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + -4.5x + x2 = 0 + 3.333333333 -4.5x + x2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 -4.5x + x2 = 3.333333333 The x term is -4.5x. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5x + 5.0625 + x2 = 3.333333333 + 5.0625 Reorder the terms: 5.0625 + -4.5x + x2 = 3.333333333 + 5.0625 Combine like terms: 3.333333333 + 5.0625 = 8.395833333 5.0625 + -4.5x + x2 = 8.395833333 Factor a perfect square on the left side: (x + -2.25)(x + -2.25) = 8.395833333 Calculate the square root of the right side: 2.897556442 Break this problem into two subproblems by setting (x + -2.25) equal to 2.897556442 and -2.897556442.Subproblem 1
x + -2.25 = 2.897556442 Simplifying x + -2.25 = 2.897556442 Reorder the terms: -2.25 + x = 2.897556442 Solving -2.25 + x = 2.897556442 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = 2.897556442 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = 2.897556442 + 2.25 x = 2.897556442 + 2.25 Combine like terms: 2.897556442 + 2.25 = 5.147556442 x = 5.147556442 Simplifying x = 5.147556442Subproblem 2
x + -2.25 = -2.897556442 Simplifying x + -2.25 = -2.897556442 Reorder the terms: -2.25 + x = -2.897556442 Solving -2.25 + x = -2.897556442 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = -2.897556442 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = -2.897556442 + 2.25 x = -2.897556442 + 2.25 Combine like terms: -2.897556442 + 2.25 = -0.647556442 x = -0.647556442 Simplifying x = -0.647556442Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.147556442, -0.647556442}Solution
x = {0, 5.147556442, -0.647556442}
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