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Simplifying (-1 + 14x)(12x + 17) = 90 Reorder the terms: (-1 + 14x)(17 + 12x) = 90 Multiply (-1 + 14x) * (17 + 12x) (-1(17 + 12x) + 14x * (17 + 12x)) = 90 ((17 * -1 + 12x * -1) + 14x * (17 + 12x)) = 90 ((-17 + -12x) + 14x * (17 + 12x)) = 90 (-17 + -12x + (17 * 14x + 12x * 14x)) = 90 (-17 + -12x + (238x + 168x2)) = 90 Combine like terms: -12x + 238x = 226x (-17 + 226x + 168x2) = 90 Solving -17 + 226x + 168x2 = 90 Solving for variable 'x'. Reorder the terms: -17 + -90 + 226x + 168x2 = 90 + -90 Combine like terms: -17 + -90 = -107 -107 + 226x + 168x2 = 90 + -90 Combine like terms: 90 + -90 = 0 -107 + 226x + 168x2 = 0 Begin completing the square. Divide all terms by 168 the coefficient of the squared term: Divide each side by '168'. -0.6369047619 + 1.345238095x + x2 = 0 Move the constant term to the right: Add '0.6369047619' to each side of the equation. -0.6369047619 + 1.345238095x + 0.6369047619 + x2 = 0 + 0.6369047619 Reorder the terms: -0.6369047619 + 0.6369047619 + 1.345238095x + x2 = 0 + 0.6369047619 Combine like terms: -0.6369047619 + 0.6369047619 = 0.0000000000 0.0000000000 + 1.345238095x + x2 = 0 + 0.6369047619 1.345238095x + x2 = 0 + 0.6369047619 Combine like terms: 0 + 0.6369047619 = 0.6369047619 1.345238095x + x2 = 0.6369047619 The x term is 1.345238095x. Take half its coefficient (0.6726190475). Square it (0.4524163831) and add it to both sides. Add '0.4524163831' to each side of the equation. 1.345238095x + 0.4524163831 + x2 = 0.6369047619 + 0.4524163831 Reorder the terms: 0.4524163831 + 1.345238095x + x2 = 0.6369047619 + 0.4524163831 Combine like terms: 0.6369047619 + 0.4524163831 = 1.089321145 0.4524163831 + 1.345238095x + x2 = 1.089321145 Factor a perfect square on the left side: (x + 0.6726190475)(x + 0.6726190475) = 1.089321145 Calculate the square root of the right side: 1.043705488 Break this problem into two subproblems by setting (x + 0.6726190475) equal to 1.043705488 and -1.043705488.Subproblem 1
x + 0.6726190475 = 1.043705488 Simplifying x + 0.6726190475 = 1.043705488 Reorder the terms: 0.6726190475 + x = 1.043705488 Solving 0.6726190475 + x = 1.043705488 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6726190475' to each side of the equation. 0.6726190475 + -0.6726190475 + x = 1.043705488 + -0.6726190475 Combine like terms: 0.6726190475 + -0.6726190475 = 0.0000000000 0.0000000000 + x = 1.043705488 + -0.6726190475 x = 1.043705488 + -0.6726190475 Combine like terms: 1.043705488 + -0.6726190475 = 0.3710864405 x = 0.3710864405 Simplifying x = 0.3710864405Subproblem 2
x + 0.6726190475 = -1.043705488 Simplifying x + 0.6726190475 = -1.043705488 Reorder the terms: 0.6726190475 + x = -1.043705488 Solving 0.6726190475 + x = -1.043705488 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6726190475' to each side of the equation. 0.6726190475 + -0.6726190475 + x = -1.043705488 + -0.6726190475 Combine like terms: 0.6726190475 + -0.6726190475 = 0.0000000000 0.0000000000 + x = -1.043705488 + -0.6726190475 x = -1.043705488 + -0.6726190475 Combine like terms: -1.043705488 + -0.6726190475 = -1.7163245355 x = -1.7163245355 Simplifying x = -1.7163245355Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.3710864405, -1.7163245355}
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