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Simplifying (7x + 19)(2x + -1) = 180 Reorder the terms: (19 + 7x)(2x + -1) = 180 Reorder the terms: (19 + 7x)(-1 + 2x) = 180 Multiply (19 + 7x) * (-1 + 2x) (19(-1 + 2x) + 7x * (-1 + 2x)) = 180 ((-1 * 19 + 2x * 19) + 7x * (-1 + 2x)) = 180 ((-19 + 38x) + 7x * (-1 + 2x)) = 180 (-19 + 38x + (-1 * 7x + 2x * 7x)) = 180 (-19 + 38x + (-7x + 14x2)) = 180 Combine like terms: 38x + -7x = 31x (-19 + 31x + 14x2) = 180 Solving -19 + 31x + 14x2 = 180 Solving for variable 'x'. Reorder the terms: -19 + -180 + 31x + 14x2 = 180 + -180 Combine like terms: -19 + -180 = -199 -199 + 31x + 14x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -199 + 31x + 14x2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -14.21428571 + 2.214285714x + x2 = 0 Move the constant term to the right: Add '14.21428571' to each side of the equation. -14.21428571 + 2.214285714x + 14.21428571 + x2 = 0 + 14.21428571 Reorder the terms: -14.21428571 + 14.21428571 + 2.214285714x + x2 = 0 + 14.21428571 Combine like terms: -14.21428571 + 14.21428571 = 0.00000000 0.00000000 + 2.214285714x + x2 = 0 + 14.21428571 2.214285714x + x2 = 0 + 14.21428571 Combine like terms: 0 + 14.21428571 = 14.21428571 2.214285714x + x2 = 14.21428571 The x term is 2.214285714x. Take half its coefficient (1.107142857). Square it (1.225765306) and add it to both sides. Add '1.225765306' to each side of the equation. 2.214285714x + 1.225765306 + x2 = 14.21428571 + 1.225765306 Reorder the terms: 1.225765306 + 2.214285714x + x2 = 14.21428571 + 1.225765306 Combine like terms: 14.21428571 + 1.225765306 = 15.440051016 1.225765306 + 2.214285714x + x2 = 15.440051016 Factor a perfect square on the left side: (x + 1.107142857)(x + 1.107142857) = 15.440051016 Calculate the square root of the right side: 3.929383032 Break this problem into two subproblems by setting (x + 1.107142857) equal to 3.929383032 and -3.929383032.Subproblem 1
x + 1.107142857 = 3.929383032 Simplifying x + 1.107142857 = 3.929383032 Reorder the terms: 1.107142857 + x = 3.929383032 Solving 1.107142857 + x = 3.929383032 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.107142857' to each side of the equation. 1.107142857 + -1.107142857 + x = 3.929383032 + -1.107142857 Combine like terms: 1.107142857 + -1.107142857 = 0.000000000 0.000000000 + x = 3.929383032 + -1.107142857 x = 3.929383032 + -1.107142857 Combine like terms: 3.929383032 + -1.107142857 = 2.822240175 x = 2.822240175 Simplifying x = 2.822240175Subproblem 2
x + 1.107142857 = -3.929383032 Simplifying x + 1.107142857 = -3.929383032 Reorder the terms: 1.107142857 + x = -3.929383032 Solving 1.107142857 + x = -3.929383032 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.107142857' to each side of the equation. 1.107142857 + -1.107142857 + x = -3.929383032 + -1.107142857 Combine like terms: 1.107142857 + -1.107142857 = 0.000000000 0.000000000 + x = -3.929383032 + -1.107142857 x = -3.929383032 + -1.107142857 Combine like terms: -3.929383032 + -1.107142857 = -5.036525889 x = -5.036525889 Simplifying x = -5.036525889Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.822240175, -5.036525889}
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