2(3a-7)(5a-6)=180

Solution for 2(3a-7)(5a-6)=180 equation:

Simplifying
2(3a + -7)(5a + -6) = 180

Reorder the terms:
2(-7 + 3a)(5a + -6) = 180

Reorder the terms:
2(-7 + 3a)(-6 + 5a) = 180

Multiply (-7 + 3a) * (-6 + 5a)
2(-7(-6 + 5a) + 3a * (-6 + 5a)) = 180
2((-6 * -7 + 5a * -7) + 3a * (-6 + 5a)) = 180
2((42 + -35a) + 3a * (-6 + 5a)) = 180
2(42 + -35a + (-6 * 3a + 5a * 3a)) = 180
2(42 + -35a + (-18a + 15a2)) = 180

Combine like terms: -35a + -18a = -53a
2(42 + -53a + 15a2) = 180
(42 * 2 + -53a * 2 + 15a2 * 2) = 180
(84 + -106a + 30a2) = 180

Solving
84 + -106a + 30a2 = 180

Solving for variable 'a'.

Reorder the terms:
84 + -180 + -106a + 30a2 = 180 + -180

Combine like terms: 84 + -180 = -96
-96 + -106a + 30a2 = 180 + -180

Combine like terms: 180 + -180 = 0
-96 + -106a + 30a2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-48 + -53a + 15a2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-48 + -53a + 15a2)' equal to zero and attempt to solve:

Simplifying
-48 + -53a + 15a2 = 0

Solving
-48 + -53a + 15a2 = 0

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

Divide each side by '15'.
-3.2 + -3.533333333a + a2 = 0

Move the constant term to the right:

Add '3.2' to each side of the equation.
-3.2 + -3.533333333a + 3.2 + a2 = 0 + 3.2

Reorder the terms:
-3.2 + 3.2 + -3.533333333a + a2 = 0 + 3.2

Combine like terms: -3.2 + 3.2 = 0.0
0.0 + -3.533333333a + a2 = 0 + 3.2
-3.533333333a + a2 = 0 + 3.2

Combine like terms: 0 + 3.2 = 3.2
-3.533333333a + a2 = 3.2

The a term is -3.533333333a.  Take half its coefficient (-1.766666667).
Square it (3.121111112) and add it to both sides.

Add '3.121111112' to each side of the equation.
-3.533333333a + 3.121111112 + a2 = 3.2 + 3.121111112

Reorder the terms:
3.121111112 + -3.533333333a + a2 = 3.2 + 3.121111112

Combine like terms: 3.2 + 3.121111112 = 6.321111112
3.121111112 + -3.533333333a + a2 = 6.321111112

Factor a perfect square on the left side:
(a + -1.766666667)(a + -1.766666667) = 6.321111112

Calculate the square root of the right side: 2.514181997

Break this problem into two subproblems by setting 
(a + -1.766666667) equal to 2.514181997 and -2.514181997.

Subproblem 1
a + -1.766666667 = 2.514181997

Simplifying
a + -1.766666667 = 2.514181997

Reorder the terms:
-1.766666667 + a = 2.514181997

Solving
-1.766666667 + a = 2.514181997

Solving for variable 'a'.

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

Add '1.766666667' to each side of the equation.
-1.766666667 + 1.766666667 + a = 2.514181997 + 1.766666667

Combine like terms: -1.766666667 + 1.766666667 = 0.000000000
0.000000000 + a = 2.514181997 + 1.766666667
a = 2.514181997 + 1.766666667

Combine like terms: 2.514181997 + 1.766666667 = 4.280848664
a = 4.280848664

Simplifying
a = 4.280848664

Subproblem 2
a + -1.766666667 = -2.514181997

Simplifying
a + -1.766666667 = -2.514181997

Reorder the terms:
-1.766666667 + a = -2.514181997

Solving
-1.766666667 + a = -2.514181997

Solving for variable 'a'.

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

Add '1.766666667' to each side of the equation.
-1.766666667 + 1.766666667 + a = -2.514181997 + 1.766666667

Combine like terms: -1.766666667 + 1.766666667 = 0.000000000
0.000000000 + a = -2.514181997 + 1.766666667
a = -2.514181997 + 1.766666667

Combine like terms: -2.514181997 + 1.766666667 = -0.74751533
a = -0.74751533

Simplifying
a = -0.74751533

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {4.280848664, -0.74751533}
Solution
a = {4.280848664, -0.74751533}