(x+109)(x+89)=180

Solution for (x+109)(x+89)=180 equation:

Simplifying
(x + 109)(x + 89) = 180

Reorder the terms:
(109 + x)(x + 89) = 180

Reorder the terms:
(109 + x)(89 + x) = 180

Multiply (109 + x) * (89 + x)
(109(89 + x) + x(89 + x)) = 180
((89 * 109 + x * 109) + x(89 + x)) = 180
((9701 + 109x) + x(89 + x)) = 180
(9701 + 109x + (89 * x + x * x)) = 180
(9701 + 109x + (89x + x2)) = 180

Combine like terms: 109x + 89x = 198x
(9701 + 198x + x2) = 180

Solving
9701 + 198x + x2 = 180

Solving for variable 'x'.

Reorder the terms:
9701 + -180 + 198x + x2 = 180 + -180

Combine like terms: 9701 + -180 = 9521
9521 + 198x + x2 = 180 + -180

Combine like terms: 180 + -180 = 0
9521 + 198x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-9521' to each side of the equation.
9521 + 198x + -9521 + x2 = 0 + -9521

Reorder the terms:
9521 + -9521 + 198x + x2 = 0 + -9521

Combine like terms: 9521 + -9521 = 0
0 + 198x + x2 = 0 + -9521
198x + x2 = 0 + -9521

Combine like terms: 0 + -9521 = -9521
198x + x2 = -9521

The x term is 198x.  Take half its coefficient (99).
Square it (9801) and add it to both sides.

Add '9801' to each side of the equation.
198x + 9801 + x2 = -9521 + 9801

Reorder the terms:
9801 + 198x + x2 = -9521 + 9801

Combine like terms: -9521 + 9801 = 280
9801 + 198x + x2 = 280

Factor a perfect square on the left side:
(x + 99)(x + 99) = 280

Calculate the square root of the right side: 16.733200531

Break this problem into two subproblems by setting 
(x + 99) equal to 16.733200531 and -16.733200531.

Subproblem 1
x + 99 = 16.733200531

Simplifying
x + 99 = 16.733200531

Reorder the terms:
99 + x = 16.733200531

Solving
99 + x = 16.733200531

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-99' to each side of the equation.
99 + -99 + x = 16.733200531 + -99

Combine like terms: 99 + -99 = 0
0 + x = 16.733200531 + -99
x = 16.733200531 + -99

Combine like terms: 16.733200531 + -99 = -82.266799469
x = -82.266799469

Simplifying
x = -82.266799469

Subproblem 2
x + 99 = -16.733200531

Simplifying
x + 99 = -16.733200531

Reorder the terms:
99 + x = -16.733200531

Solving
99 + x = -16.733200531

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-99' to each side of the equation.
99 + -99 + x = -16.733200531 + -99

Combine like terms: 99 + -99 = 0
0 + x = -16.733200531 + -99
x = -16.733200531 + -99

Combine like terms: -16.733200531 + -99 = -115.733200531
x = -115.733200531

Simplifying
x = -115.733200531

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-82.266799469, -115.733200531}