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Simplifying -24x3 + -12x2 + 13x = 0 Reorder the terms: 13x + -12x2 + -24x3 = 0 Solving 13x + -12x2 + -24x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x'. x(13 + -12x + -24x2) = 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 '(13 + -12x + -24x2)' equal to zero and attempt to solve: Simplifying 13 + -12x + -24x2 = 0 Solving 13 + -12x + -24x2 = 0 Begin completing the square. Divide all terms by -24 the coefficient of the squared term: Divide each side by '-24'. -0.5416666667 + 0.5x + x2 = 0 Move the constant term to the right: Add '0.5416666667' to each side of the equation. -0.5416666667 + 0.5x + 0.5416666667 + x2 = 0 + 0.5416666667 Reorder the terms: -0.5416666667 + 0.5416666667 + 0.5x + x2 = 0 + 0.5416666667 Combine like terms: -0.5416666667 + 0.5416666667 = 0.0000000000 0.0000000000 + 0.5x + x2 = 0 + 0.5416666667 0.5x + x2 = 0 + 0.5416666667 Combine like terms: 0 + 0.5416666667 = 0.5416666667 0.5x + x2 = 0.5416666667 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 0.5416666667 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 0.5416666667 + 0.0625 Combine like terms: 0.5416666667 + 0.0625 = 0.6041666667 0.0625 + 0.5x + x2 = 0.6041666667 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 0.6041666667 Calculate the square root of the right side: 0.777281588 Break this problem into two subproblems by setting (x + 0.25) equal to 0.777281588 and -0.777281588.Subproblem 1
x + 0.25 = 0.777281588 Simplifying x + 0.25 = 0.777281588 Reorder the terms: 0.25 + x = 0.777281588 Solving 0.25 + x = 0.777281588 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 0.777281588 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.777281588 + -0.25 x = 0.777281588 + -0.25 Combine like terms: 0.777281588 + -0.25 = 0.527281588 x = 0.527281588 Simplifying x = 0.527281588Subproblem 2
x + 0.25 = -0.777281588 Simplifying x + 0.25 = -0.777281588 Reorder the terms: 0.25 + x = -0.777281588 Solving 0.25 + x = -0.777281588 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -0.777281588 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.777281588 + -0.25 x = -0.777281588 + -0.25 Combine like terms: -0.777281588 + -0.25 = -1.027281588 x = -1.027281588 Simplifying x = -1.027281588Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.527281588, -1.027281588}Solution
x = {0, 0.527281588, -1.027281588}
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