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Simplifying (8x + 13)(12x + -7) = 90 Reorder the terms: (13 + 8x)(12x + -7) = 90 Reorder the terms: (13 + 8x)(-7 + 12x) = 90 Multiply (13 + 8x) * (-7 + 12x) (13(-7 + 12x) + 8x * (-7 + 12x)) = 90 ((-7 * 13 + 12x * 13) + 8x * (-7 + 12x)) = 90 ((-91 + 156x) + 8x * (-7 + 12x)) = 90 (-91 + 156x + (-7 * 8x + 12x * 8x)) = 90 (-91 + 156x + (-56x + 96x2)) = 90 Combine like terms: 156x + -56x = 100x (-91 + 100x + 96x2) = 90 Solving -91 + 100x + 96x2 = 90 Solving for variable 'x'. Reorder the terms: -91 + -90 + 100x + 96x2 = 90 + -90 Combine like terms: -91 + -90 = -181 -181 + 100x + 96x2 = 90 + -90 Combine like terms: 90 + -90 = 0 -181 + 100x + 96x2 = 0 Begin completing the square. Divide all terms by 96 the coefficient of the squared term: Divide each side by '96'. -1.885416667 + 1.041666667x + x2 = 0 Move the constant term to the right: Add '1.885416667' to each side of the equation. -1.885416667 + 1.041666667x + 1.885416667 + x2 = 0 + 1.885416667 Reorder the terms: -1.885416667 + 1.885416667 + 1.041666667x + x2 = 0 + 1.885416667 Combine like terms: -1.885416667 + 1.885416667 = 0.000000000 0.000000000 + 1.041666667x + x2 = 0 + 1.885416667 1.041666667x + x2 = 0 + 1.885416667 Combine like terms: 0 + 1.885416667 = 1.885416667 1.041666667x + x2 = 1.885416667 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 = 1.885416667 + 0.2712673613 Reorder the terms: 0.2712673613 + 1.041666667x + x2 = 1.885416667 + 0.2712673613 Combine like terms: 1.885416667 + 0.2712673613 = 2.1566840283 0.2712673613 + 1.041666667x + x2 = 2.1566840283 Factor a perfect square on the left side: (x + 0.5208333335)(x + 0.5208333335) = 2.1566840283 Calculate the square root of the right side: 1.468565296 Break this problem into two subproblems by setting (x + 0.5208333335) equal to 1.468565296 and -1.468565296.Subproblem 1
x + 0.5208333335 = 1.468565296 Simplifying x + 0.5208333335 = 1.468565296 Reorder the terms: 0.5208333335 + x = 1.468565296 Solving 0.5208333335 + x = 1.468565296 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 = 1.468565296 + -0.5208333335 Combine like terms: 0.5208333335 + -0.5208333335 = 0.0000000000 0.0000000000 + x = 1.468565296 + -0.5208333335 x = 1.468565296 + -0.5208333335 Combine like terms: 1.468565296 + -0.5208333335 = 0.9477319625 x = 0.9477319625 Simplifying x = 0.9477319625Subproblem 2
x + 0.5208333335 = -1.468565296 Simplifying x + 0.5208333335 = -1.468565296 Reorder the terms: 0.5208333335 + x = -1.468565296 Solving 0.5208333335 + x = -1.468565296 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 = -1.468565296 + -0.5208333335 Combine like terms: 0.5208333335 + -0.5208333335 = 0.0000000000 0.0000000000 + x = -1.468565296 + -0.5208333335 x = -1.468565296 + -0.5208333335 Combine like terms: -1.468565296 + -0.5208333335 = -1.9893986295 x = -1.9893986295 Simplifying x = -1.9893986295Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.9477319625, -1.9893986295}
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