4a^2-(23a+4)(34a-6)=12

Solution for 4a^2-(23a+4)(34a-6)=12 equation:

Simplifying
4a2 + -1(23a + 4)(34a + -6) = 12

Reorder the terms:
4a2 + -1(4 + 23a)(34a + -6) = 12

Reorder the terms:
4a2 + -1(4 + 23a)(-6 + 34a) = 12

Multiply (4 + 23a) * (-6 + 34a)
4a2 + -1(4(-6 + 34a) + 23a * (-6 + 34a)) = 12
4a2 + -1((-6 * 4 + 34a * 4) + 23a * (-6 + 34a)) = 12
4a2 + -1((-24 + 136a) + 23a * (-6 + 34a)) = 12
4a2 + -1(-24 + 136a + (-6 * 23a + 34a * 23a)) = 12
4a2 + -1(-24 + 136a + (-138a + 782a2)) = 12

Combine like terms: 136a + -138a = -2a
4a2 + -1(-24 + -2a + 782a2) = 12
4a2 + (-24 * -1 + -2a * -1 + 782a2 * -1) = 12
4a2 + (24 + 2a + -782a2) = 12

Reorder the terms:
24 + 2a + 4a2 + -782a2 = 12

Combine like terms: 4a2 + -782a2 = -778a2
24 + 2a + -778a2 = 12

Solving
24 + 2a + -778a2 = 12

Solving for variable 'a'.

Reorder the terms:
24 + -12 + 2a + -778a2 = 12 + -12

Combine like terms: 24 + -12 = 12
12 + 2a + -778a2 = 12 + -12

Combine like terms: 12 + -12 = 0
12 + 2a + -778a2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(6 + a + -389a2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(6 + a + -389a2)' equal to zero and attempt to solve:

Simplifying
6 + a + -389a2 = 0

Solving
6 + a + -389a2 = 0

Begin completing the square.  Divide all terms by
-389 the coefficient of the squared term: 

Divide each side by '-389'.
-0.01542416452 + -0.002570694087a + a2 = 0

Move the constant term to the right:

Add '0.01542416452' to each side of the equation.
-0.01542416452 + -0.002570694087a + 0.01542416452 + a2 = 0 + 0.01542416452

Reorder the terms:
-0.01542416452 + 0.01542416452 + -0.002570694087a + a2 = 0 + 0.01542416452

Combine like terms: -0.01542416452 + 0.01542416452 = 0.00000000000
0.00000000000 + -0.002570694087a + a2 = 0 + 0.01542416452
-0.002570694087a + a2 = 0 + 0.01542416452

Combine like terms: 0 + 0.01542416452 = 0.01542416452
-0.002570694087a + a2 = 0.01542416452

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

Add '0.25' to each side of the equation.
-0.002570694087a + 0.25 + a2 = 0.01542416452 + 0.25

Reorder the terms:
0.25 + -0.002570694087a + a2 = 0.01542416452 + 0.25

Combine like terms: 0.01542416452 + 0.25 = 0.26542416452
0.25 + -0.002570694087a + a2 = 0.26542416452

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 0.26542416452

Calculate the square root of the right side: 0.515193327

Break this problem into two subproblems by setting 
(a + 0.5) equal to 0.515193327 and -0.515193327.

Subproblem 1
a + 0.5 = 0.515193327

Simplifying
a + 0.5 = 0.515193327

Reorder the terms:
0.5 + a = 0.515193327

Solving
0.5 + a = 0.515193327

Solving for variable 'a'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + a = 0.515193327 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + a = 0.515193327 + -0.5
a = 0.515193327 + -0.5

Combine like terms: 0.515193327 + -0.5 = 0.015193327
a = 0.015193327

Simplifying
a = 0.015193327

Subproblem 2
a + 0.5 = -0.515193327

Simplifying
a + 0.5 = -0.515193327

Reorder the terms:
0.5 + a = -0.515193327

Solving
0.5 + a = -0.515193327

Solving for variable 'a'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + a = -0.515193327 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + a = -0.515193327 + -0.5
a = -0.515193327 + -0.5

Combine like terms: -0.515193327 + -0.5 = -1.015193327
a = -1.015193327

Simplifying
a = -1.015193327

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.015193327, -1.015193327}
Solution
a = {0.015193327, -1.015193327}