10a(5a+3)=15

Solution for 10a(5a+3)=15 equation:

Simplifying
10a(5a + 3) = 15

Reorder the terms:
10a(3 + 5a) = 15
(3 * 10a + 5a * 10a) = 15
(30a + 50a2) = 15

Solving
30a + 50a2 = 15

Solving for variable 'a'.

Reorder the terms:
-15 + 30a + 50a2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + 30a + 50a2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-3 + 6a + 10a2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(-3 + 6a + 10a2)' equal to zero and attempt to solve:

Simplifying
-3 + 6a + 10a2 = 0

Solving
-3 + 6a + 10a2 = 0

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

Divide each side by '10'.
-0.3 + 0.6a + a2 = 0.0

Move the constant term to the right:

Add '0.3' to each side of the equation.
-0.3 + 0.6a + 0.3 + a2 = 0.0 + 0.3

Reorder the terms:
-0.3 + 0.3 + 0.6a + a2 = 0.0 + 0.3

Combine like terms: -0.3 + 0.3 = 0.0
0.0 + 0.6a + a2 = 0.0 + 0.3
0.6a + a2 = 0.0 + 0.3

Combine like terms: 0.0 + 0.3 = 0.3
0.6a + a2 = 0.3

The a term is 0.6a.  Take half its coefficient (0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
0.6a + 0.09 + a2 = 0.3 + 0.09

Reorder the terms:
0.09 + 0.6a + a2 = 0.3 + 0.09

Combine like terms: 0.3 + 0.09 = 0.39
0.09 + 0.6a + a2 = 0.39

Factor a perfect square on the left side:
(a + 0.3)(a + 0.3) = 0.39

Calculate the square root of the right side: 0.6244998

Break this problem into two subproblems by setting 
(a + 0.3) equal to 0.6244998 and -0.6244998.

Subproblem 1
a + 0.3 = 0.6244998

Simplifying
a + 0.3 = 0.6244998

Reorder the terms:
0.3 + a = 0.6244998

Solving
0.3 + a = 0.6244998

Solving for variable 'a'.

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

Add '-0.3' to each side of the equation.
0.3 + -0.3 + a = 0.6244998 + -0.3

Combine like terms: 0.3 + -0.3 = 0.0
0.0 + a = 0.6244998 + -0.3
a = 0.6244998 + -0.3

Combine like terms: 0.6244998 + -0.3 = 0.3244998
a = 0.3244998

Simplifying
a = 0.3244998

Subproblem 2
a + 0.3 = -0.6244998

Simplifying
a + 0.3 = -0.6244998

Reorder the terms:
0.3 + a = -0.6244998

Solving
0.3 + a = -0.6244998

Solving for variable 'a'.

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

Add '-0.3' to each side of the equation.
0.3 + -0.3 + a = -0.6244998 + -0.3

Combine like terms: 0.3 + -0.3 = 0.0
0.0 + a = -0.6244998 + -0.3
a = -0.6244998 + -0.3

Combine like terms: -0.6244998 + -0.3 = -0.9244998
a = -0.9244998

Simplifying
a = -0.9244998

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.3244998, -0.9244998}
Solution
a = {0.3244998, -0.9244998}