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Simplifying (5x + 23)(7x + 13) = 180 Reorder the terms: (23 + 5x)(7x + 13) = 180 Reorder the terms: (23 + 5x)(13 + 7x) = 180 Multiply (23 + 5x) * (13 + 7x) (23(13 + 7x) + 5x * (13 + 7x)) = 180 ((13 * 23 + 7x * 23) + 5x * (13 + 7x)) = 180 ((299 + 161x) + 5x * (13 + 7x)) = 180 (299 + 161x + (13 * 5x + 7x * 5x)) = 180 (299 + 161x + (65x + 35x2)) = 180 Combine like terms: 161x + 65x = 226x (299 + 226x + 35x2) = 180 Solving 299 + 226x + 35x2 = 180 Solving for variable 'x'. Reorder the terms: 299 + -180 + 226x + 35x2 = 180 + -180 Combine like terms: 299 + -180 = 119 119 + 226x + 35x2 = 180 + -180 Combine like terms: 180 + -180 = 0 119 + 226x + 35x2 = 0 Begin completing the square. Divide all terms by 35 the coefficient of the squared term: Divide each side by '35'. 3.4 + 6.457142857x + x2 = 0 Move the constant term to the right: Add '-3.4' to each side of the equation. 3.4 + 6.457142857x + -3.4 + x2 = 0 + -3.4 Reorder the terms: 3.4 + -3.4 + 6.457142857x + x2 = 0 + -3.4 Combine like terms: 3.4 + -3.4 = 0.0 0.0 + 6.457142857x + x2 = 0 + -3.4 6.457142857x + x2 = 0 + -3.4 Combine like terms: 0 + -3.4 = -3.4 6.457142857x + x2 = -3.4 The x term is 6.457142857x. Take half its coefficient (3.228571429). Square it (10.42367347) and add it to both sides. Add '10.42367347' to each side of the equation. 6.457142857x + 10.42367347 + x2 = -3.4 + 10.42367347 Reorder the terms: 10.42367347 + 6.457142857x + x2 = -3.4 + 10.42367347 Combine like terms: -3.4 + 10.42367347 = 7.02367347 10.42367347 + 6.457142857x + x2 = 7.02367347 Factor a perfect square on the left side: (x + 3.228571429)(x + 3.228571429) = 7.02367347 Calculate the square root of the right side: 2.6502214 Break this problem into two subproblems by setting (x + 3.228571429) equal to 2.6502214 and -2.6502214.Subproblem 1
x + 3.228571429 = 2.6502214 Simplifying x + 3.228571429 = 2.6502214 Reorder the terms: 3.228571429 + x = 2.6502214 Solving 3.228571429 + x = 2.6502214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.228571429' to each side of the equation. 3.228571429 + -3.228571429 + x = 2.6502214 + -3.228571429 Combine like terms: 3.228571429 + -3.228571429 = 0.000000000 0.000000000 + x = 2.6502214 + -3.228571429 x = 2.6502214 + -3.228571429 Combine like terms: 2.6502214 + -3.228571429 = -0.578350029 x = -0.578350029 Simplifying x = -0.578350029Subproblem 2
x + 3.228571429 = -2.6502214 Simplifying x + 3.228571429 = -2.6502214 Reorder the terms: 3.228571429 + x = -2.6502214 Solving 3.228571429 + x = -2.6502214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.228571429' to each side of the equation. 3.228571429 + -3.228571429 + x = -2.6502214 + -3.228571429 Combine like terms: 3.228571429 + -3.228571429 = 0.000000000 0.000000000 + x = -2.6502214 + -3.228571429 x = -2.6502214 + -3.228571429 Combine like terms: -2.6502214 + -3.228571429 = -5.878792829 x = -5.878792829 Simplifying x = -5.878792829Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.578350029, -5.878792829}
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