4a(2a+50)=180

Solution for 4a(2a+50)=180 equation:

Simplifying
4a(2a + 50) = 180

Reorder the terms:
4a(50 + 2a) = 180
(50 * 4a + 2a * 4a) = 180
(200a + 8a2) = 180

Solving
200a + 8a2 = 180

Solving for variable 'a'.

Reorder the terms:
-180 + 200a + 8a2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 200a + 8a2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-45 + 50a + 2a2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-45 + 50a + 2a2)' equal to zero and attempt to solve:

Simplifying
-45 + 50a + 2a2 = 0

Solving
-45 + 50a + 2a2 = 0

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

Divide each side by '2'.
-22.5 + 25a + a2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + 25a + 22.5 + a2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + 25a + a2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + 25a + a2 = 0 + 22.5
25a + a2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
25a + a2 = 22.5

The a term is 25a.  Take half its coefficient (12.5).
Square it (156.25) and add it to both sides.

Add '156.25' to each side of the equation.
25a + 156.25 + a2 = 22.5 + 156.25

Reorder the terms:
156.25 + 25a + a2 = 22.5 + 156.25

Combine like terms: 22.5 + 156.25 = 178.75
156.25 + 25a + a2 = 178.75

Factor a perfect square on the left side:
(a + 12.5)(a + 12.5) = 178.75

Calculate the square root of the right side: 13.369741957

Break this problem into two subproblems by setting 
(a + 12.5) equal to 13.369741957 and -13.369741957.

Subproblem 1
a + 12.5 = 13.369741957

Simplifying
a + 12.5 = 13.369741957

Reorder the terms:
12.5 + a = 13.369741957

Solving
12.5 + a = 13.369741957

Solving for variable 'a'.

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

Add '-12.5' to each side of the equation.
12.5 + -12.5 + a = 13.369741957 + -12.5

Combine like terms: 12.5 + -12.5 = 0.0
0.0 + a = 13.369741957 + -12.5
a = 13.369741957 + -12.5

Combine like terms: 13.369741957 + -12.5 = 0.869741957
a = 0.869741957

Simplifying
a = 0.869741957

Subproblem 2
a + 12.5 = -13.369741957

Simplifying
a + 12.5 = -13.369741957

Reorder the terms:
12.5 + a = -13.369741957

Solving
12.5 + a = -13.369741957

Solving for variable 'a'.

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

Add '-12.5' to each side of the equation.
12.5 + -12.5 + a = -13.369741957 + -12.5

Combine like terms: 12.5 + -12.5 = 0.0
0.0 + a = -13.369741957 + -12.5
a = -13.369741957 + -12.5

Combine like terms: -13.369741957 + -12.5 = -25.869741957
a = -25.869741957

Simplifying
a = -25.869741957

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.869741957, -25.869741957}
Solution
a = {0.869741957, -25.869741957}