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Simplifying 24x2 + 16x + -6 = 0 Reorder the terms: -6 + 16x + 24x2 = 0 Solving -6 + 16x + 24x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 8x + 12x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + 8x + 12x2)' equal to zero and attempt to solve: Simplifying -3 + 8x + 12x2 = 0 Solving -3 + 8x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.25 + 0.6666666667x + x2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 0.6666666667x + 0.25 + x2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 0.6666666667x + x2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 0.6666666667x + x2 = 0 + 0.25 0.6666666667x + x2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 0.6666666667x + x2 = 0.25 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 = 0.25 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667x + x2 = 0.25 + 0.1111111112 Combine like terms: 0.25 + 0.1111111112 = 0.3611111112 0.1111111112 + 0.6666666667x + x2 = 0.3611111112 Factor a perfect square on the left side: (x + 0.3333333334)(x + 0.3333333334) = 0.3611111112 Calculate the square root of the right side: 0.600925213 Break this problem into two subproblems by setting (x + 0.3333333334) equal to 0.600925213 and -0.600925213.Subproblem 1
x + 0.3333333334 = 0.600925213 Simplifying x + 0.3333333334 = 0.600925213 Reorder the terms: 0.3333333334 + x = 0.600925213 Solving 0.3333333334 + x = 0.600925213 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 = 0.600925213 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 0.600925213 + -0.3333333334 x = 0.600925213 + -0.3333333334 Combine like terms: 0.600925213 + -0.3333333334 = 0.2675918796 x = 0.2675918796 Simplifying x = 0.2675918796Subproblem 2
x + 0.3333333334 = -0.600925213 Simplifying x + 0.3333333334 = -0.600925213 Reorder the terms: 0.3333333334 + x = -0.600925213 Solving 0.3333333334 + x = -0.600925213 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 = -0.600925213 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -0.600925213 + -0.3333333334 x = -0.600925213 + -0.3333333334 Combine like terms: -0.600925213 + -0.3333333334 = -0.9342585464 x = -0.9342585464 Simplifying x = -0.9342585464Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.2675918796, -0.9342585464}Solution
x = {0.2675918796, -0.9342585464}
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