16a(2a+3)=8+20

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Solution for 16a(2a+3)=8+20 equation: 


Simplifying
16a(2a + 3) = 8 + 20

Reorder the terms:
16a(3 + 2a) = 8 + 20
(3 * 16a + 2a * 16a) = 8 + 20
(48a + 32a2) = 8 + 20

Combine like terms: 8 + 20 = 28
48a + 32a2 = 28

Solving
48a + 32a2 = 28

Solving for variable 'a'.

Reorder the terms:
-28 + 48a + 32a2 = 28 + -28

Combine like terms: 28 + -28 = 0
-28 + 48a + 32a2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-7 + 12a + 8a2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-7 + 12a + 8a2)' equal to zero and attempt to solve:

Simplifying
-7 + 12a + 8a2 = 0

Solving
-7 + 12a + 8a2 = 0

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

Divide each side by '8'.
-0.875 + 1.5a + a2 = 0

Move the constant term to the right:

Add '0.875' to each side of the equation.
-0.875 + 1.5a + 0.875 + a2 = 0 + 0.875

Reorder the terms:
-0.875 + 0.875 + 1.5a + a2 = 0 + 0.875

Combine like terms: -0.875 + 0.875 = 0.000
0.000 + 1.5a + a2 = 0 + 0.875
1.5a + a2 = 0 + 0.875

Combine like terms: 0 + 0.875 = 0.875
1.5a + a2 = 0.875

The a term is 1.5a.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5a + 0.5625 + a2 = 0.875 + 0.5625

Reorder the terms:
0.5625 + 1.5a + a2 = 0.875 + 0.5625

Combine like terms: 0.875 + 0.5625 = 1.4375
0.5625 + 1.5a + a2 = 1.4375

Factor a perfect square on the left side:
(a + 0.75)(a + 0.75) = 1.4375

Calculate the square root of the right side: 1.198957881

Break this problem into two subproblems by setting 
(a + 0.75) equal to 1.198957881 and -1.198957881.


Subproblem 1
a + 0.75 = 1.198957881

Simplifying
a + 0.75 = 1.198957881

Reorder the terms:
0.75 + a = 1.198957881

Solving
0.75 + a = 1.198957881

Solving for variable 'a'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + a = 1.198957881 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + a = 1.198957881 + -0.75
a = 1.198957881 + -0.75

Combine like terms: 1.198957881 + -0.75 = 0.448957881
a = 0.448957881

Simplifying
a = 0.448957881


Subproblem 2
a + 0.75 = -1.198957881

Simplifying
a + 0.75 = -1.198957881

Reorder the terms:
0.75 + a = -1.198957881

Solving
0.75 + a = -1.198957881

Solving for variable 'a'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + a = -1.198957881 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + a = -1.198957881 + -0.75
a = -1.198957881 + -0.75

Combine like terms: -1.198957881 + -0.75 = -1.948957881
a = -1.948957881

Simplifying
a = -1.948957881


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.448957881, -1.948957881}
Solution
a = {0.448957881, -1.948957881}

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