a+6=a(a+3)

Solution for a+6=a(a+3) equation:

Simplifying
a + 6 = a(a + 3)

Reorder the terms:
6 + a = a(a + 3)

Reorder the terms:
6 + a = a(3 + a)
6 + a = (3 * a + a * a)
6 + a = (3a + a2)

Solving
6 + a = 3a + a2

Solving for variable 'a'.

Combine like terms: a + -3a = -2a
6 + -2a + -1a2 = 3a + a2 + -3a + -1a2

Reorder the terms:
6 + -2a + -1a2 = 3a + -3a + a2 + -1a2

Combine like terms: 3a + -3a = 0
6 + -2a + -1a2 = 0 + a2 + -1a2
6 + -2a + -1a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
6 + -2a + -1a2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-6 + 2a + a2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 2a + 6 + a2 = 0 + 6

Reorder the terms:
-6 + 6 + 2a + a2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 2a + a2 = 0 + 6
2a + a2 = 0 + 6

Combine like terms: 0 + 6 = 6
2a + a2 = 6

The a term is 2a.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2a + 1 + a2 = 6 + 1

Reorder the terms:
1 + 2a + a2 = 6 + 1

Combine like terms: 6 + 1 = 7
1 + 2a + a2 = 7

Factor a perfect square on the left side:
(a + 1)(a + 1) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(a + 1) equal to 2.645751311 and -2.645751311.

Subproblem 1
a + 1 = 2.645751311

Simplifying
a + 1 = 2.645751311

Reorder the terms:
1 + a = 2.645751311

Solving
1 + a = 2.645751311

Solving for variable 'a'.

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

Add '-1' to each side of the equation.
1 + -1 + a = 2.645751311 + -1

Combine like terms: 1 + -1 = 0
0 + a = 2.645751311 + -1
a = 2.645751311 + -1

Combine like terms: 2.645751311 + -1 = 1.645751311
a = 1.645751311

Simplifying
a = 1.645751311

Subproblem 2
a + 1 = -2.645751311

Simplifying
a + 1 = -2.645751311

Reorder the terms:
1 + a = -2.645751311

Solving
1 + a = -2.645751311

Solving for variable 'a'.

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

Add '-1' to each side of the equation.
1 + -1 + a = -2.645751311 + -1

Combine like terms: 1 + -1 = 0
0 + a = -2.645751311 + -1
a = -2.645751311 + -1

Combine like terms: -2.645751311 + -1 = -3.645751311
a = -3.645751311

Simplifying
a = -3.645751311

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {1.645751311, -3.645751311}