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Simplifying (8x + 13)(12x + -7) = 180 Reorder the terms: (13 + 8x)(12x + -7) = 180 Reorder the terms: (13 + 8x)(-7 + 12x) = 180 Multiply (13 + 8x) * (-7 + 12x) (13(-7 + 12x) + 8x * (-7 + 12x)) = 180 ((-7 * 13 + 12x * 13) + 8x * (-7 + 12x)) = 180 ((-91 + 156x) + 8x * (-7 + 12x)) = 180 (-91 + 156x + (-7 * 8x + 12x * 8x)) = 180 (-91 + 156x + (-56x + 96x2)) = 180 Combine like terms: 156x + -56x = 100x (-91 + 100x + 96x2) = 180 Solving -91 + 100x + 96x2 = 180 Solving for variable 'x'. Reorder the terms: -91 + -180 + 100x + 96x2 = 180 + -180 Combine like terms: -91 + -180 = -271 -271 + 100x + 96x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -271 + 100x + 96x2 = 0 Begin completing the square. Divide all terms by 96 the coefficient of the squared term: Divide each side by '96'. -2.822916667 + 1.041666667x + x2 = 0 Move the constant term to the right: Add '2.822916667' to each side of the equation. -2.822916667 + 1.041666667x + 2.822916667 + x2 = 0 + 2.822916667 Reorder the terms: -2.822916667 + 2.822916667 + 1.041666667x + x2 = 0 + 2.822916667 Combine like terms: -2.822916667 + 2.822916667 = 0.000000000 0.000000000 + 1.041666667x + x2 = 0 + 2.822916667 1.041666667x + x2 = 0 + 2.822916667 Combine like terms: 0 + 2.822916667 = 2.822916667 1.041666667x + x2 = 2.822916667 The x term is 1.041666667x. Take half its coefficient (0.5208333335). Square it (0.2712673613) and add it to both sides. Add '0.2712673613' to each side of the equation. 1.041666667x + 0.2712673613 + x2 = 2.822916667 + 0.2712673613 Reorder the terms: 0.2712673613 + 1.041666667x + x2 = 2.822916667 + 0.2712673613 Combine like terms: 2.822916667 + 0.2712673613 = 3.0941840283 0.2712673613 + 1.041666667x + x2 = 3.0941840283 Factor a perfect square on the left side: (x + 0.5208333335)(x + 0.5208333335) = 3.0941840283 Calculate the square root of the right side: 1.759029286 Break this problem into two subproblems by setting (x + 0.5208333335) equal to 1.759029286 and -1.759029286.Subproblem 1
x + 0.5208333335 = 1.759029286 Simplifying x + 0.5208333335 = 1.759029286 Reorder the terms: 0.5208333335 + x = 1.759029286 Solving 0.5208333335 + x = 1.759029286 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5208333335' to each side of the equation. 0.5208333335 + -0.5208333335 + x = 1.759029286 + -0.5208333335 Combine like terms: 0.5208333335 + -0.5208333335 = 0.0000000000 0.0000000000 + x = 1.759029286 + -0.5208333335 x = 1.759029286 + -0.5208333335 Combine like terms: 1.759029286 + -0.5208333335 = 1.2381959525 x = 1.2381959525 Simplifying x = 1.2381959525Subproblem 2
x + 0.5208333335 = -1.759029286 Simplifying x + 0.5208333335 = -1.759029286 Reorder the terms: 0.5208333335 + x = -1.759029286 Solving 0.5208333335 + x = -1.759029286 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5208333335' to each side of the equation. 0.5208333335 + -0.5208333335 + x = -1.759029286 + -0.5208333335 Combine like terms: 0.5208333335 + -0.5208333335 = 0.0000000000 0.0000000000 + x = -1.759029286 + -0.5208333335 x = -1.759029286 + -0.5208333335 Combine like terms: -1.759029286 + -0.5208333335 = -2.2798626195 x = -2.2798626195 Simplifying x = -2.2798626195Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.2381959525, -2.2798626195}
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