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Simplifying (6x + 7)(11x + -2) = 180 Reorder the terms: (7 + 6x)(11x + -2) = 180 Reorder the terms: (7 + 6x)(-2 + 11x) = 180 Multiply (7 + 6x) * (-2 + 11x) (7(-2 + 11x) + 6x * (-2 + 11x)) = 180 ((-2 * 7 + 11x * 7) + 6x * (-2 + 11x)) = 180 ((-14 + 77x) + 6x * (-2 + 11x)) = 180 (-14 + 77x + (-2 * 6x + 11x * 6x)) = 180 (-14 + 77x + (-12x + 66x2)) = 180 Combine like terms: 77x + -12x = 65x (-14 + 65x + 66x2) = 180 Solving -14 + 65x + 66x2 = 180 Solving for variable 'x'. Reorder the terms: -14 + -180 + 65x + 66x2 = 180 + -180 Combine like terms: -14 + -180 = -194 -194 + 65x + 66x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -194 + 65x + 66x2 = 0 Begin completing the square. Divide all terms by 66 the coefficient of the squared term: Divide each side by '66'. -2.939393939 + 0.9848484848x + x2 = 0 Move the constant term to the right: Add '2.939393939' to each side of the equation. -2.939393939 + 0.9848484848x + 2.939393939 + x2 = 0 + 2.939393939 Reorder the terms: -2.939393939 + 2.939393939 + 0.9848484848x + x2 = 0 + 2.939393939 Combine like terms: -2.939393939 + 2.939393939 = 0.000000000 0.000000000 + 0.9848484848x + x2 = 0 + 2.939393939 0.9848484848x + x2 = 0 + 2.939393939 Combine like terms: 0 + 2.939393939 = 2.939393939 0.9848484848x + x2 = 2.939393939 The x term is 0.9848484848x. Take half its coefficient (0.4924242424). Square it (0.2424816345) and add it to both sides. Add '0.2424816345' to each side of the equation. 0.9848484848x + 0.2424816345 + x2 = 2.939393939 + 0.2424816345 Reorder the terms: 0.2424816345 + 0.9848484848x + x2 = 2.939393939 + 0.2424816345 Combine like terms: 2.939393939 + 0.2424816345 = 3.1818755735 0.2424816345 + 0.9848484848x + x2 = 3.1818755735 Factor a perfect square on the left side: (x + 0.4924242424)(x + 0.4924242424) = 3.1818755735 Calculate the square root of the right side: 1.783781257 Break this problem into two subproblems by setting (x + 0.4924242424) equal to 1.783781257 and -1.783781257.Subproblem 1
x + 0.4924242424 = 1.783781257 Simplifying x + 0.4924242424 = 1.783781257 Reorder the terms: 0.4924242424 + x = 1.783781257 Solving 0.4924242424 + x = 1.783781257 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4924242424' to each side of the equation. 0.4924242424 + -0.4924242424 + x = 1.783781257 + -0.4924242424 Combine like terms: 0.4924242424 + -0.4924242424 = 0.0000000000 0.0000000000 + x = 1.783781257 + -0.4924242424 x = 1.783781257 + -0.4924242424 Combine like terms: 1.783781257 + -0.4924242424 = 1.2913570146 x = 1.2913570146 Simplifying x = 1.2913570146Subproblem 2
x + 0.4924242424 = -1.783781257 Simplifying x + 0.4924242424 = -1.783781257 Reorder the terms: 0.4924242424 + x = -1.783781257 Solving 0.4924242424 + x = -1.783781257 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4924242424' to each side of the equation. 0.4924242424 + -0.4924242424 + x = -1.783781257 + -0.4924242424 Combine like terms: 0.4924242424 + -0.4924242424 = 0.0000000000 0.0000000000 + x = -1.783781257 + -0.4924242424 x = -1.783781257 + -0.4924242424 Combine like terms: -1.783781257 + -0.4924242424 = -2.2762054994 x = -2.2762054994 Simplifying x = -2.2762054994Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.2913570146, -2.2762054994}
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