14a(4a+5)=25(4a+6)

Solution for 14a(4a+5)=25(4a+6) equation:

Simplifying
14a(4a + 5) = 25(4a + 6)

Reorder the terms:
14a(5 + 4a) = 25(4a + 6)
(5 * 14a + 4a * 14a) = 25(4a + 6)
(70a + 56a2) = 25(4a + 6)

Reorder the terms:
70a + 56a2 = 25(6 + 4a)
70a + 56a2 = (6 * 25 + 4a * 25)
70a + 56a2 = (150 + 100a)

Solving
70a + 56a2 = 150 + 100a

Solving for variable 'a'.

Reorder the terms:
-150 + 70a + -100a + 56a2 = 150 + 100a + -150 + -100a

Combine like terms: 70a + -100a = -30a
-150 + -30a + 56a2 = 150 + 100a + -150 + -100a

Reorder the terms:
-150 + -30a + 56a2 = 150 + -150 + 100a + -100a

Combine like terms: 150 + -150 = 0
-150 + -30a + 56a2 = 0 + 100a + -100a
-150 + -30a + 56a2 = 100a + -100a

Combine like terms: 100a + -100a = 0
-150 + -30a + 56a2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-75 + -15a + 28a2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-75 + -15a + 28a2)' equal to zero and attempt to solve:

Simplifying
-75 + -15a + 28a2 = 0

Solving
-75 + -15a + 28a2 = 0

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

Divide each side by '28'.
-2.678571429 + -0.5357142857a + a2 = 0

Move the constant term to the right:

Add '2.678571429' to each side of the equation.
-2.678571429 + -0.5357142857a + 2.678571429 + a2 = 0 + 2.678571429

Reorder the terms:
-2.678571429 + 2.678571429 + -0.5357142857a + a2 = 0 + 2.678571429

Combine like terms: -2.678571429 + 2.678571429 = 0.000000000
0.000000000 + -0.5357142857a + a2 = 0 + 2.678571429
-0.5357142857a + a2 = 0 + 2.678571429

Combine like terms: 0 + 2.678571429 = 2.678571429
-0.5357142857a + a2 = 2.678571429

The a term is -0.5357142857a.  Take half its coefficient (-0.2678571429).
Square it (0.07174744900) and add it to both sides.

Add '0.07174744900' to each side of the equation.
-0.5357142857a + 0.07174744900 + a2 = 2.678571429 + 0.07174744900

Reorder the terms:
0.07174744900 + -0.5357142857a + a2 = 2.678571429 + 0.07174744900

Combine like terms: 2.678571429 + 0.07174744900 = 2.750318878
0.07174744900 + -0.5357142857a + a2 = 2.750318878

Factor a perfect square on the left side:
(a + -0.2678571429)(a + -0.2678571429) = 2.750318878

Calculate the square root of the right side: 1.658408538

Break this problem into two subproblems by setting 
(a + -0.2678571429) equal to 1.658408538 and -1.658408538.

Subproblem 1
a + -0.2678571429 = 1.658408538

Simplifying
a + -0.2678571429 = 1.658408538

Reorder the terms:
-0.2678571429 + a = 1.658408538

Solving
-0.2678571429 + a = 1.658408538

Solving for variable 'a'.

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

Add '0.2678571429' to each side of the equation.
-0.2678571429 + 0.2678571429 + a = 1.658408538 + 0.2678571429

Combine like terms: -0.2678571429 + 0.2678571429 = 0.0000000000
0.0000000000 + a = 1.658408538 + 0.2678571429
a = 1.658408538 + 0.2678571429

Combine like terms: 1.658408538 + 0.2678571429 = 1.9262656809
a = 1.9262656809

Simplifying
a = 1.9262656809

Subproblem 2
a + -0.2678571429 = -1.658408538

Simplifying
a + -0.2678571429 = -1.658408538

Reorder the terms:
-0.2678571429 + a = -1.658408538

Solving
-0.2678571429 + a = -1.658408538

Solving for variable 'a'.

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

Add '0.2678571429' to each side of the equation.
-0.2678571429 + 0.2678571429 + a = -1.658408538 + 0.2678571429

Combine like terms: -0.2678571429 + 0.2678571429 = 0.0000000000
0.0000000000 + a = -1.658408538 + 0.2678571429
a = -1.658408538 + 0.2678571429

Combine like terms: -1.658408538 + 0.2678571429 = -1.3905513951
a = -1.3905513951

Simplifying
a = -1.3905513951

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {1.9262656809, -1.3905513951}
Solution
a = {1.9262656809, -1.3905513951}