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Simplifying (2x + -1)(3x + 8) = 1 Reorder the terms: (-1 + 2x)(3x + 8) = 1 Reorder the terms: (-1 + 2x)(8 + 3x) = 1 Multiply (-1 + 2x) * (8 + 3x) (-1(8 + 3x) + 2x * (8 + 3x)) = 1 ((8 * -1 + 3x * -1) + 2x * (8 + 3x)) = 1 ((-8 + -3x) + 2x * (8 + 3x)) = 1 (-8 + -3x + (8 * 2x + 3x * 2x)) = 1 (-8 + -3x + (16x + 6x2)) = 1 Combine like terms: -3x + 16x = 13x (-8 + 13x + 6x2) = 1 Solving -8 + 13x + 6x2 = 1 Solving for variable 'x'. Reorder the terms: -8 + -1 + 13x + 6x2 = 1 + -1 Combine like terms: -8 + -1 = -9 -9 + 13x + 6x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -9 + 13x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.5 + 2.166666667x + x2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 2.166666667x + 1.5 + x2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 2.166666667x + x2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 2.166666667x + x2 = 0 + 1.5 2.166666667x + x2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 2.166666667x + x2 = 1.5 The x term is 2.166666667x. Take half its coefficient (1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. 2.166666667x + 1.173611113 + x2 = 1.5 + 1.173611113 Reorder the terms: 1.173611113 + 2.166666667x + x2 = 1.5 + 1.173611113 Combine like terms: 1.5 + 1.173611113 = 2.673611113 1.173611113 + 2.166666667x + x2 = 2.673611113 Factor a perfect square on the left side: (x + 1.083333334)(x + 1.083333334) = 2.673611113 Calculate the square root of the right side: 1.635118073 Break this problem into two subproblems by setting (x + 1.083333334) equal to 1.635118073 and -1.635118073.Subproblem 1
x + 1.083333334 = 1.635118073 Simplifying x + 1.083333334 = 1.635118073 Reorder the terms: 1.083333334 + x = 1.635118073 Solving 1.083333334 + x = 1.635118073 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + x = 1.635118073 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + x = 1.635118073 + -1.083333334 x = 1.635118073 + -1.083333334 Combine like terms: 1.635118073 + -1.083333334 = 0.551784739 x = 0.551784739 Simplifying x = 0.551784739Subproblem 2
x + 1.083333334 = -1.635118073 Simplifying x + 1.083333334 = -1.635118073 Reorder the terms: 1.083333334 + x = -1.635118073 Solving 1.083333334 + x = -1.635118073 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + x = -1.635118073 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + x = -1.635118073 + -1.083333334 x = -1.635118073 + -1.083333334 Combine like terms: -1.635118073 + -1.083333334 = -2.718451407 x = -2.718451407 Simplifying x = -2.718451407Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.551784739, -2.718451407}
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