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Simplifying 24x2 + 82x + 40 = 0 Reorder the terms: 40 + 82x + 24x2 = 0 Solving 40 + 82x + 24x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(20 + 41x + 12x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(20 + 41x + 12x2)' equal to zero and attempt to solve: Simplifying 20 + 41x + 12x2 = 0 Solving 20 + 41x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 1.666666667 + 3.416666667x + x2 = 0 Move the constant term to the right: Add '-1.666666667' to each side of the equation. 1.666666667 + 3.416666667x + -1.666666667 + x2 = 0 + -1.666666667 Reorder the terms: 1.666666667 + -1.666666667 + 3.416666667x + x2 = 0 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + 3.416666667x + x2 = 0 + -1.666666667 3.416666667x + x2 = 0 + -1.666666667 Combine like terms: 0 + -1.666666667 = -1.666666667 3.416666667x + x2 = -1.666666667 The x term is 3.416666667x. Take half its coefficient (1.708333334). Square it (2.918402780) and add it to both sides. Add '2.918402780' to each side of the equation. 3.416666667x + 2.918402780 + x2 = -1.666666667 + 2.918402780 Reorder the terms: 2.918402780 + 3.416666667x + x2 = -1.666666667 + 2.918402780 Combine like terms: -1.666666667 + 2.918402780 = 1.251736113 2.918402780 + 3.416666667x + x2 = 1.251736113 Factor a perfect square on the left side: (x + 1.708333334)(x + 1.708333334) = 1.251736113 Calculate the square root of the right side: 1.118810133 Break this problem into two subproblems by setting (x + 1.708333334) equal to 1.118810133 and -1.118810133.Subproblem 1
x + 1.708333334 = 1.118810133 Simplifying x + 1.708333334 = 1.118810133 Reorder the terms: 1.708333334 + x = 1.118810133 Solving 1.708333334 + x = 1.118810133 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.708333334' to each side of the equation. 1.708333334 + -1.708333334 + x = 1.118810133 + -1.708333334 Combine like terms: 1.708333334 + -1.708333334 = 0.000000000 0.000000000 + x = 1.118810133 + -1.708333334 x = 1.118810133 + -1.708333334 Combine like terms: 1.118810133 + -1.708333334 = -0.589523201 x = -0.589523201 Simplifying x = -0.589523201Subproblem 2
x + 1.708333334 = -1.118810133 Simplifying x + 1.708333334 = -1.118810133 Reorder the terms: 1.708333334 + x = -1.118810133 Solving 1.708333334 + x = -1.118810133 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.708333334' to each side of the equation. 1.708333334 + -1.708333334 + x = -1.118810133 + -1.708333334 Combine like terms: 1.708333334 + -1.708333334 = 0.000000000 0.000000000 + x = -1.118810133 + -1.708333334 x = -1.118810133 + -1.708333334 Combine like terms: -1.118810133 + -1.708333334 = -2.827143467 x = -2.827143467 Simplifying x = -2.827143467Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.589523201, -2.827143467}Solution
x = {-0.589523201, -2.827143467}
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