(5x+23)(7x+13)=180

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Solution for (5x+23)(7x+13)=180 equation:


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 = -0.578350029

Subproblem 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 = -5.878792829

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.578350029, -5.878792829}

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