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Simplifying (7x + -12)(5x + 18) = 180 Reorder the terms: (-12 + 7x)(5x + 18) = 180 Reorder the terms: (-12 + 7x)(18 + 5x) = 180 Multiply (-12 + 7x) * (18 + 5x) (-12(18 + 5x) + 7x * (18 + 5x)) = 180 ((18 * -12 + 5x * -12) + 7x * (18 + 5x)) = 180 ((-216 + -60x) + 7x * (18 + 5x)) = 180 (-216 + -60x + (18 * 7x + 5x * 7x)) = 180 (-216 + -60x + (126x + 35x2)) = 180 Combine like terms: -60x + 126x = 66x (-216 + 66x + 35x2) = 180 Solving -216 + 66x + 35x2 = 180 Solving for variable 'x'. Reorder the terms: -216 + -180 + 66x + 35x2 = 180 + -180 Combine like terms: -216 + -180 = -396 -396 + 66x + 35x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -396 + 66x + 35x2 = 0 Begin completing the square. Divide all terms by 35 the coefficient of the squared term: Divide each side by '35'. -11.31428571 + 1.885714286x + x2 = 0 Move the constant term to the right: Add '11.31428571' to each side of the equation. -11.31428571 + 1.885714286x + 11.31428571 + x2 = 0 + 11.31428571 Reorder the terms: -11.31428571 + 11.31428571 + 1.885714286x + x2 = 0 + 11.31428571 Combine like terms: -11.31428571 + 11.31428571 = 0.00000000 0.00000000 + 1.885714286x + x2 = 0 + 11.31428571 1.885714286x + x2 = 0 + 11.31428571 Combine like terms: 0 + 11.31428571 = 11.31428571 1.885714286x + x2 = 11.31428571 The x term is 1.885714286x. Take half its coefficient (0.942857143). Square it (0.8889795921) and add it to both sides. Add '0.8889795921' to each side of the equation. 1.885714286x + 0.8889795921 + x2 = 11.31428571 + 0.8889795921 Reorder the terms: 0.8889795921 + 1.885714286x + x2 = 11.31428571 + 0.8889795921 Combine like terms: 11.31428571 + 0.8889795921 = 12.2032653021 0.8889795921 + 1.885714286x + x2 = 12.2032653021 Factor a perfect square on the left side: (x + 0.942857143)(x + 0.942857143) = 12.2032653021 Calculate the square root of the right side: 3.493317235 Break this problem into two subproblems by setting (x + 0.942857143) equal to 3.493317235 and -3.493317235.Subproblem 1
x + 0.942857143 = 3.493317235 Simplifying x + 0.942857143 = 3.493317235 Reorder the terms: 0.942857143 + x = 3.493317235 Solving 0.942857143 + x = 3.493317235 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.942857143' to each side of the equation. 0.942857143 + -0.942857143 + x = 3.493317235 + -0.942857143 Combine like terms: 0.942857143 + -0.942857143 = 0.000000000 0.000000000 + x = 3.493317235 + -0.942857143 x = 3.493317235 + -0.942857143 Combine like terms: 3.493317235 + -0.942857143 = 2.550460092 x = 2.550460092 Simplifying x = 2.550460092Subproblem 2
x + 0.942857143 = -3.493317235 Simplifying x + 0.942857143 = -3.493317235 Reorder the terms: 0.942857143 + x = -3.493317235 Solving 0.942857143 + x = -3.493317235 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.942857143' to each side of the equation. 0.942857143 + -0.942857143 + x = -3.493317235 + -0.942857143 Combine like terms: 0.942857143 + -0.942857143 = 0.000000000 0.000000000 + x = -3.493317235 + -0.942857143 x = -3.493317235 + -0.942857143 Combine like terms: -3.493317235 + -0.942857143 = -4.436174378 x = -4.436174378 Simplifying x = -4.436174378Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.550460092, -4.436174378}
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