3+.5a(4a+8)=9-2a

Solution for 3+.5a(4a+8)=9-2a equation:

Simplifying
3 + 0.5a(4a + 8) = 9 + -2a

Reorder the terms:
3 + 0.5a(8 + 4a) = 9 + -2a
3 + (8 * 0.5a + 4a * 0.5a) = 9 + -2a
3 + (4a + 2a2) = 9 + -2a

Solving
3 + 4a + 2a2 = 9 + -2a

Solving for variable 'a'.

Reorder the terms:
3 + -9 + 4a + 2a + 2a2 = 9 + -2a + -9 + 2a

Combine like terms: 3 + -9 = -6
-6 + 4a + 2a + 2a2 = 9 + -2a + -9 + 2a

Combine like terms: 4a + 2a = 6a
-6 + 6a + 2a2 = 9 + -2a + -9 + 2a

Reorder the terms:
-6 + 6a + 2a2 = 9 + -9 + -2a + 2a

Combine like terms: 9 + -9 = 0
-6 + 6a + 2a2 = 0 + -2a + 2a
-6 + 6a + 2a2 = -2a + 2a

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

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + 3a + a2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-3 + 3a + a2)' equal to zero and attempt to solve:

Simplifying
-3 + 3a + a2 = 0

Solving
-3 + 3a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 3a + 3 + a2 = 0 + 3

Reorder the terms:
-3 + 3 + 3a + a2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 3a + a2 = 0 + 3
3a + a2 = 0 + 3

Combine like terms: 0 + 3 = 3
3a + a2 = 3

The a term is 3a.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3a + 2.25 + a2 = 3 + 2.25

Reorder the terms:
2.25 + 3a + a2 = 3 + 2.25

Combine like terms: 3 + 2.25 = 5.25
2.25 + 3a + a2 = 5.25

Factor a perfect square on the left side:
(a + 1.5)(a + 1.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(a + 1.5) equal to 2.291287847 and -2.291287847.

Subproblem 1
a + 1.5 = 2.291287847

Simplifying
a + 1.5 = 2.291287847

Reorder the terms:
1.5 + a = 2.291287847

Solving
1.5 + a = 2.291287847

Solving for variable 'a'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + a = 2.291287847 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + a = 2.291287847 + -1.5
a = 2.291287847 + -1.5

Combine like terms: 2.291287847 + -1.5 = 0.791287847
a = 0.791287847

Simplifying
a = 0.791287847

Subproblem 2
a + 1.5 = -2.291287847

Simplifying
a + 1.5 = -2.291287847

Reorder the terms:
1.5 + a = -2.291287847

Solving
1.5 + a = -2.291287847

Solving for variable 'a'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + a = -2.291287847 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + a = -2.291287847 + -1.5
a = -2.291287847 + -1.5

Combine like terms: -2.291287847 + -1.5 = -3.791287847
a = -3.791287847

Simplifying
a = -3.791287847

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.791287847, -3.791287847}
Solution
a = {0.791287847, -3.791287847}