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Simplifying 24x3 + -16x2 = 68x Reorder the terms: -16x2 + 24x3 = 68x Solving -16x2 + 24x3 = 68x Solving for variable 'x'. Reorder the terms: -68x + -16x2 + 24x3 = 68x + -68x Combine like terms: 68x + -68x = 0 -68x + -16x2 + 24x3 = 0 Factor out the Greatest Common Factor (GCF), '4x'. 4x(-17 + -4x + 6x2) = 0 Ignore the factor 4.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 '(-17 + -4x + 6x2)' equal to zero and attempt to solve: Simplifying -17 + -4x + 6x2 = 0 Solving -17 + -4x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.833333333 + -0.6666666667x + x2 = 0 Move the constant term to the right: Add '2.833333333' to each side of the equation. -2.833333333 + -0.6666666667x + 2.833333333 + x2 = 0 + 2.833333333 Reorder the terms: -2.833333333 + 2.833333333 + -0.6666666667x + x2 = 0 + 2.833333333 Combine like terms: -2.833333333 + 2.833333333 = 0.000000000 0.000000000 + -0.6666666667x + x2 = 0 + 2.833333333 -0.6666666667x + x2 = 0 + 2.833333333 Combine like terms: 0 + 2.833333333 = 2.833333333 -0.6666666667x + x2 = 2.833333333 The x term is -0.6666666667x. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667x + 0.1111111112 + x2 = 2.833333333 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667x + x2 = 2.833333333 + 0.1111111112 Combine like terms: 2.833333333 + 0.1111111112 = 2.9444444442 0.1111111112 + -0.6666666667x + x2 = 2.9444444442 Factor a perfect square on the left side: (x + -0.3333333334)(x + -0.3333333334) = 2.9444444442 Calculate the square root of the right side: 1.715938357 Break this problem into two subproblems by setting (x + -0.3333333334) equal to 1.715938357 and -1.715938357.Subproblem 1
x + -0.3333333334 = 1.715938357 Simplifying x + -0.3333333334 = 1.715938357 Reorder the terms: -0.3333333334 + x = 1.715938357 Solving -0.3333333334 + x = 1.715938357 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = 1.715938357 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 1.715938357 + 0.3333333334 x = 1.715938357 + 0.3333333334 Combine like terms: 1.715938357 + 0.3333333334 = 2.0492716904 x = 2.0492716904 Simplifying x = 2.0492716904Subproblem 2
x + -0.3333333334 = -1.715938357 Simplifying x + -0.3333333334 = -1.715938357 Reorder the terms: -0.3333333334 + x = -1.715938357 Solving -0.3333333334 + x = -1.715938357 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = -1.715938357 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -1.715938357 + 0.3333333334 x = -1.715938357 + 0.3333333334 Combine like terms: -1.715938357 + 0.3333333334 = -1.3826050236 x = -1.3826050236 Simplifying x = -1.3826050236Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.0492716904, -1.3826050236}Solution
x = {0, 2.0492716904, -1.3826050236}
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