2(2a+3)7(4a-8)=7

Solution for 2(2a+3)7(4a-8)=7 equation:

Simplifying
2(2a + 3) * 7(4a + -8) = 7

Reorder the terms:
2(3 + 2a) * 7(4a + -8) = 7

Reorder the terms:
2(3 + 2a) * 7(-8 + 4a) = 7

Reorder the terms for easier multiplication:
2 * 7(3 + 2a)(-8 + 4a) = 7

Multiply 2 * 7
14(3 + 2a)(-8 + 4a) = 7

Multiply (3 + 2a) * (-8 + 4a)
14(3(-8 + 4a) + 2a * (-8 + 4a)) = 7
14((-8 * 3 + 4a * 3) + 2a * (-8 + 4a)) = 7
14((-24 + 12a) + 2a * (-8 + 4a)) = 7
14(-24 + 12a + (-8 * 2a + 4a * 2a)) = 7
14(-24 + 12a + (-16a + 8a2)) = 7

Combine like terms: 12a + -16a = -4a
14(-24 + -4a + 8a2) = 7
(-24 * 14 + -4a * 14 + 8a2 * 14) = 7
(-336 + -56a + 112a2) = 7

Solving
-336 + -56a + 112a2 = 7

Solving for variable 'a'.

Reorder the terms:
-336 + -7 + -56a + 112a2 = 7 + -7

Combine like terms: -336 + -7 = -343
-343 + -56a + 112a2 = 7 + -7

Combine like terms: 7 + -7 = 0
-343 + -56a + 112a2 = 0

Factor out the Greatest Common Factor (GCF), '7'.
7(-49 + -8a + 16a2) = 0

Ignore the factor 7.
Subproblem 1
Set the factor '(-49 + -8a + 16a2)' equal to zero and attempt to solve:

Simplifying
-49 + -8a + 16a2 = 0

Solving
-49 + -8a + 16a2 = 0

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

Divide each side by '16'.
-3.0625 + -0.5a + a2 = 0

Move the constant term to the right:

Add '3.0625' to each side of the equation.
-3.0625 + -0.5a + 3.0625 + a2 = 0 + 3.0625

Reorder the terms:
-3.0625 + 3.0625 + -0.5a + a2 = 0 + 3.0625

Combine like terms: -3.0625 + 3.0625 = 0.0000
0.0000 + -0.5a + a2 = 0 + 3.0625
-0.5a + a2 = 0 + 3.0625

Combine like terms: 0 + 3.0625 = 3.0625
-0.5a + a2 = 3.0625

The a term is -0.5a.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5a + 0.0625 + a2 = 3.0625 + 0.0625

Reorder the terms:
0.0625 + -0.5a + a2 = 3.0625 + 0.0625

Combine like terms: 3.0625 + 0.0625 = 3.125
0.0625 + -0.5a + a2 = 3.125

Factor a perfect square on the left side:
(a + -0.25)(a + -0.25) = 3.125

Calculate the square root of the right side: 1.767766953

Break this problem into two subproblems by setting 
(a + -0.25) equal to 1.767766953 and -1.767766953.

Subproblem 1
a + -0.25 = 1.767766953

Simplifying
a + -0.25 = 1.767766953

Reorder the terms:
-0.25 + a = 1.767766953

Solving
-0.25 + a = 1.767766953

Solving for variable 'a'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + a = 1.767766953 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + a = 1.767766953 + 0.25
a = 1.767766953 + 0.25

Combine like terms: 1.767766953 + 0.25 = 2.017766953
a = 2.017766953

Simplifying
a = 2.017766953

Subproblem 2
a + -0.25 = -1.767766953

Simplifying
a + -0.25 = -1.767766953

Reorder the terms:
-0.25 + a = -1.767766953

Solving
-0.25 + a = -1.767766953

Solving for variable 'a'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + a = -1.767766953 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + a = -1.767766953 + 0.25
a = -1.767766953 + 0.25

Combine like terms: -1.767766953 + 0.25 = -1.517766953
a = -1.517766953

Simplifying
a = -1.517766953

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {2.017766953, -1.517766953}
Solution
a = {2.017766953, -1.517766953}