(5a+12)(9a-12)=180

Solution for (5a+12)(9a-12)=180 equation:

Simplifying
(5a + 12)(9a + -12) = 180

Reorder the terms:
(12 + 5a)(9a + -12) = 180

Reorder the terms:
(12 + 5a)(-12 + 9a) = 180

Multiply (12 + 5a) * (-12 + 9a)
(12(-12 + 9a) + 5a * (-12 + 9a)) = 180
((-12 * 12 + 9a * 12) + 5a * (-12 + 9a)) = 180
((-144 + 108a) + 5a * (-12 + 9a)) = 180
(-144 + 108a + (-12 * 5a + 9a * 5a)) = 180
(-144 + 108a + (-60a + 45a2)) = 180

Combine like terms: 108a + -60a = 48a
(-144 + 48a + 45a2) = 180

Solving
-144 + 48a + 45a2 = 180

Solving for variable 'a'.

Reorder the terms:
-144 + -180 + 48a + 45a2 = 180 + -180

Combine like terms: -144 + -180 = -324
-324 + 48a + 45a2 = 180 + -180

Combine like terms: 180 + -180 = 0
-324 + 48a + 45a2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-108 + 16a + 15a2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-108 + 16a + 15a2)' equal to zero and attempt to solve:

Simplifying
-108 + 16a + 15a2 = 0

Solving
-108 + 16a + 15a2 = 0

Begin completing the square.  Divide all terms by
15 the coefficient of the squared term: 

Divide each side by '15'.
-7.2 + 1.066666667a + a2 = 0

Move the constant term to the right:

Add '7.2' to each side of the equation.
-7.2 + 1.066666667a + 7.2 + a2 = 0 + 7.2

Reorder the terms:
-7.2 + 7.2 + 1.066666667a + a2 = 0 + 7.2

Combine like terms: -7.2 + 7.2 = 0.0
0.0 + 1.066666667a + a2 = 0 + 7.2
1.066666667a + a2 = 0 + 7.2

Combine like terms: 0 + 7.2 = 7.2
1.066666667a + a2 = 7.2

The a term is 1.066666667a.  Take half its coefficient (0.5333333335).
Square it (0.2844444446) and add it to both sides.

Add '0.2844444446' to each side of the equation.
1.066666667a + 0.2844444446 + a2 = 7.2 + 0.2844444446

Reorder the terms:
0.2844444446 + 1.066666667a + a2 = 7.2 + 0.2844444446

Combine like terms: 7.2 + 0.2844444446 = 7.4844444446
0.2844444446 + 1.066666667a + a2 = 7.4844444446

Factor a perfect square on the left side:
(a + 0.5333333335)(a + 0.5333333335) = 7.4844444446

Calculate the square root of the right side: 2.735771271

Break this problem into two subproblems by setting 
(a + 0.5333333335) equal to 2.735771271 and -2.735771271.

Subproblem 1
a + 0.5333333335 = 2.735771271

Simplifying
a + 0.5333333335 = 2.735771271

Reorder the terms:
0.5333333335 + a = 2.735771271

Solving
0.5333333335 + a = 2.735771271

Solving for variable 'a'.

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

Add '-0.5333333335' to each side of the equation.
0.5333333335 + -0.5333333335 + a = 2.735771271 + -0.5333333335

Combine like terms: 0.5333333335 + -0.5333333335 = 0.0000000000
0.0000000000 + a = 2.735771271 + -0.5333333335
a = 2.735771271 + -0.5333333335

Combine like terms: 2.735771271 + -0.5333333335 = 2.2024379375
a = 2.2024379375

Simplifying
a = 2.2024379375

Subproblem 2
a + 0.5333333335 = -2.735771271

Simplifying
a + 0.5333333335 = -2.735771271

Reorder the terms:
0.5333333335 + a = -2.735771271

Solving
0.5333333335 + a = -2.735771271

Solving for variable 'a'.

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

Add '-0.5333333335' to each side of the equation.
0.5333333335 + -0.5333333335 + a = -2.735771271 + -0.5333333335

Combine like terms: 0.5333333335 + -0.5333333335 = 0.0000000000
0.0000000000 + a = -2.735771271 + -0.5333333335
a = -2.735771271 + -0.5333333335

Combine like terms: -2.735771271 + -0.5333333335 = -3.2691046045
a = -3.2691046045

Simplifying
a = -3.2691046045

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {2.2024379375, -3.2691046045}
Solution
a = {2.2024379375, -3.2691046045}