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Simplifying (3x + 13)(3x + -5) = -24 Reorder the terms: (13 + 3x)(3x + -5) = -24 Reorder the terms: (13 + 3x)(-5 + 3x) = -24 Multiply (13 + 3x) * (-5 + 3x) (13(-5 + 3x) + 3x * (-5 + 3x)) = -24 ((-5 * 13 + 3x * 13) + 3x * (-5 + 3x)) = -24 ((-65 + 39x) + 3x * (-5 + 3x)) = -24 (-65 + 39x + (-5 * 3x + 3x * 3x)) = -24 (-65 + 39x + (-15x + 9x2)) = -24 Combine like terms: 39x + -15x = 24x (-65 + 24x + 9x2) = -24 Solving -65 + 24x + 9x2 = -24 Solving for variable 'x'. Reorder the terms: -65 + 24 + 24x + 9x2 = -24 + 24 Combine like terms: -65 + 24 = -41 -41 + 24x + 9x2 = -24 + 24 Combine like terms: -24 + 24 = 0 -41 + 24x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -4.555555556 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '4.555555556' to each side of the equation. -4.555555556 + 2.666666667x + 4.555555556 + x2 = 0 + 4.555555556 Reorder the terms: -4.555555556 + 4.555555556 + 2.666666667x + x2 = 0 + 4.555555556 Combine like terms: -4.555555556 + 4.555555556 = 0.000000000 0.000000000 + 2.666666667x + x2 = 0 + 4.555555556 2.666666667x + x2 = 0 + 4.555555556 Combine like terms: 0 + 4.555555556 = 4.555555556 2.666666667x + x2 = 4.555555556 The x term is 2.666666667x. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667x + 1.777777780 + x2 = 4.555555556 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = 4.555555556 + 1.777777780 Combine like terms: 4.555555556 + 1.777777780 = 6.333333336 1.777777780 + 2.666666667x + x2 = 6.333333336 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 6.333333336 Calculate the square root of the right side: 2.516611479 Break this problem into two subproblems by setting (x + 1.333333334) equal to 2.516611479 and -2.516611479.Subproblem 1
x + 1.333333334 = 2.516611479 Simplifying x + 1.333333334 = 2.516611479 Reorder the terms: 1.333333334 + x = 2.516611479 Solving 1.333333334 + x = 2.516611479 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 2.516611479 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 2.516611479 + -1.333333334 x = 2.516611479 + -1.333333334 Combine like terms: 2.516611479 + -1.333333334 = 1.183278145 x = 1.183278145 Simplifying x = 1.183278145Subproblem 2
x + 1.333333334 = -2.516611479 Simplifying x + 1.333333334 = -2.516611479 Reorder the terms: 1.333333334 + x = -2.516611479 Solving 1.333333334 + x = -2.516611479 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -2.516611479 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -2.516611479 + -1.333333334 x = -2.516611479 + -1.333333334 Combine like terms: -2.516611479 + -1.333333334 = -3.849944813 x = -3.849944813 Simplifying x = -3.849944813Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.183278145, -3.849944813}
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