3(a-5)=6a(3a+15)

Solution for 3(a-5)=6a(3a+15) equation:

Simplifying
3(a + -5) = 6a(3a + 15)

Reorder the terms:
3(-5 + a) = 6a(3a + 15)
(-5 * 3 + a * 3) = 6a(3a + 15)
(-15 + 3a) = 6a(3a + 15)

Reorder the terms:
-15 + 3a = 6a(15 + 3a)
-15 + 3a = (15 * 6a + 3a * 6a)
-15 + 3a = (90a + 18a2)

Solving
-15 + 3a = 90a + 18a2

Solving for variable 'a'.

Combine like terms: 3a + -90a = -87a
-15 + -87a + -18a2 = 90a + 18a2 + -90a + -18a2

Reorder the terms:
-15 + -87a + -18a2 = 90a + -90a + 18a2 + -18a2

Combine like terms: 90a + -90a = 0
-15 + -87a + -18a2 = 0 + 18a2 + -18a2
-15 + -87a + -18a2 = 18a2 + -18a2

Combine like terms: 18a2 + -18a2 = 0
-15 + -87a + -18a2 = 0

Factor out the Greatest Common Factor (GCF), '-3'.
-3(5 + 29a + 6a2) = 0

Ignore the factor -3.
Subproblem 1
Set the factor '(5 + 29a + 6a2)' equal to zero and attempt to solve:

Simplifying
5 + 29a + 6a2 = 0

Solving
5 + 29a + 6a2 = 0

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

Divide each side by '6'.
0.8333333333 + 4.833333333a + a2 = 0

Move the constant term to the right:

Add '-0.8333333333' to each side of the equation.
0.8333333333 + 4.833333333a + -0.8333333333 + a2 = 0 + -0.8333333333

Reorder the terms:
0.8333333333 + -0.8333333333 + 4.833333333a + a2 = 0 + -0.8333333333

Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000
0.0000000000 + 4.833333333a + a2 = 0 + -0.8333333333
4.833333333a + a2 = 0 + -0.8333333333

Combine like terms: 0 + -0.8333333333 = -0.8333333333
4.833333333a + a2 = -0.8333333333

The a term is 4.833333333a.  Take half its coefficient (2.416666667).
Square it (5.840277779) and add it to both sides.

Add '5.840277779' to each side of the equation.
4.833333333a + 5.840277779 + a2 = -0.8333333333 + 5.840277779

Reorder the terms:
5.840277779 + 4.833333333a + a2 = -0.8333333333 + 5.840277779

Combine like terms: -0.8333333333 + 5.840277779 = 5.0069444457
5.840277779 + 4.833333333a + a2 = 5.0069444457

Factor a perfect square on the left side:
(a + 2.416666667)(a + 2.416666667) = 5.0069444457

Calculate the square root of the right side: 2.237620264

Break this problem into two subproblems by setting 
(a + 2.416666667) equal to 2.237620264 and -2.237620264.

Subproblem 1
a + 2.416666667 = 2.237620264

Simplifying
a + 2.416666667 = 2.237620264

Reorder the terms:
2.416666667 + a = 2.237620264

Solving
2.416666667 + a = 2.237620264

Solving for variable 'a'.

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

Add '-2.416666667' to each side of the equation.
2.416666667 + -2.416666667 + a = 2.237620264 + -2.416666667

Combine like terms: 2.416666667 + -2.416666667 = 0.000000000
0.000000000 + a = 2.237620264 + -2.416666667
a = 2.237620264 + -2.416666667

Combine like terms: 2.237620264 + -2.416666667 = -0.179046403
a = -0.179046403

Simplifying
a = -0.179046403

Subproblem 2
a + 2.416666667 = -2.237620264

Simplifying
a + 2.416666667 = -2.237620264

Reorder the terms:
2.416666667 + a = -2.237620264

Solving
2.416666667 + a = -2.237620264

Solving for variable 'a'.

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

Add '-2.416666667' to each side of the equation.
2.416666667 + -2.416666667 + a = -2.237620264 + -2.416666667

Combine like terms: 2.416666667 + -2.416666667 = 0.000000000
0.000000000 + a = -2.237620264 + -2.416666667
a = -2.237620264 + -2.416666667

Combine like terms: -2.237620264 + -2.416666667 = -4.654286931
a = -4.654286931

Simplifying
a = -4.654286931

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-0.179046403, -4.654286931}
Solution
a = {-0.179046403, -4.654286931}