3a(a+4)+9=(a-3)(a-7)

Solution for 3a(a+4)+9=(a-3)(a-7) equation:

Simplifying
3a(a + 4) + 9 = (a + -3)(a + -7)

Reorder the terms:
3a(4 + a) + 9 = (a + -3)(a + -7)
(4 * 3a + a * 3a) + 9 = (a + -3)(a + -7)
(12a + 3a2) + 9 = (a + -3)(a + -7)

Reorder the terms:
9 + 12a + 3a2 = (a + -3)(a + -7)

Reorder the terms:
9 + 12a + 3a2 = (-3 + a)(a + -7)

Reorder the terms:
9 + 12a + 3a2 = (-3 + a)(-7 + a)

Multiply (-3 + a) * (-7 + a)
9 + 12a + 3a2 = (-3(-7 + a) + a(-7 + a))
9 + 12a + 3a2 = ((-7 * -3 + a * -3) + a(-7 + a))
9 + 12a + 3a2 = ((21 + -3a) + a(-7 + a))
9 + 12a + 3a2 = (21 + -3a + (-7 * a + a * a))
9 + 12a + 3a2 = (21 + -3a + (-7a + a2))

Combine like terms: -3a + -7a = -10a
9 + 12a + 3a2 = (21 + -10a + a2)

Solving
9 + 12a + 3a2 = 21 + -10a + a2

Solving for variable 'a'.

Reorder the terms:
9 + -21 + 12a + 10a + 3a2 + -1a2 = 21 + -10a + a2 + -21 + 10a + -1a2

Combine like terms: 9 + -21 = -12
-12 + 12a + 10a + 3a2 + -1a2 = 21 + -10a + a2 + -21 + 10a + -1a2

Combine like terms: 12a + 10a = 22a
-12 + 22a + 3a2 + -1a2 = 21 + -10a + a2 + -21 + 10a + -1a2

Combine like terms: 3a2 + -1a2 = 2a2
-12 + 22a + 2a2 = 21 + -10a + a2 + -21 + 10a + -1a2

Reorder the terms:
-12 + 22a + 2a2 = 21 + -21 + -10a + 10a + a2 + -1a2

Combine like terms: 21 + -21 = 0
-12 + 22a + 2a2 = 0 + -10a + 10a + a2 + -1a2
-12 + 22a + 2a2 = -10a + 10a + a2 + -1a2

Combine like terms: -10a + 10a = 0
-12 + 22a + 2a2 = 0 + a2 + -1a2
-12 + 22a + 2a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
-12 + 22a + 2a2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-6 + 11a + a2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-6 + 11a + a2)' equal to zero and attempt to solve:

Simplifying
-6 + 11a + a2 = 0

Solving
-6 + 11a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 11a + 6 + a2 = 0 + 6

Reorder the terms:
-6 + 6 + 11a + a2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 11a + a2 = 0 + 6
11a + a2 = 0 + 6

Combine like terms: 0 + 6 = 6
11a + a2 = 6

The a term is 11a.  Take half its coefficient (5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
11a + 30.25 + a2 = 6 + 30.25

Reorder the terms:
30.25 + 11a + a2 = 6 + 30.25

Combine like terms: 6 + 30.25 = 36.25
30.25 + 11a + a2 = 36.25

Factor a perfect square on the left side:
(a + 5.5)(a + 5.5) = 36.25

Calculate the square root of the right side: 6.020797289

Break this problem into two subproblems by setting 
(a + 5.5) equal to 6.020797289 and -6.020797289.

Subproblem 1
a + 5.5 = 6.020797289

Simplifying
a + 5.5 = 6.020797289

Reorder the terms:
5.5 + a = 6.020797289

Solving
5.5 + a = 6.020797289

Solving for variable 'a'.

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

Add '-5.5' to each side of the equation.
5.5 + -5.5 + a = 6.020797289 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + a = 6.020797289 + -5.5
a = 6.020797289 + -5.5

Combine like terms: 6.020797289 + -5.5 = 0.520797289
a = 0.520797289

Simplifying
a = 0.520797289

Subproblem 2
a + 5.5 = -6.020797289

Simplifying
a + 5.5 = -6.020797289

Reorder the terms:
5.5 + a = -6.020797289

Solving
5.5 + a = -6.020797289

Solving for variable 'a'.

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

Add '-5.5' to each side of the equation.
5.5 + -5.5 + a = -6.020797289 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + a = -6.020797289 + -5.5
a = -6.020797289 + -5.5

Combine like terms: -6.020797289 + -5.5 = -11.520797289
a = -11.520797289

Simplifying
a = -11.520797289

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.520797289, -11.520797289}
Solution
a = {0.520797289, -11.520797289}