6(24x^2+24x+1)=0

Solution for 6(24x^2+24x+1)=0 equation:

Simplifying
6(24x2 + 24x + 1) = 0

Reorder the terms:
6(1 + 24x + 24x2) = 0
(1 * 6 + 24x * 6 + 24x2 * 6) = 0
(6 + 144x + 144x2) = 0

Solving
6 + 144x + 144x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6'.
6(1 + 24x + 24x2) = 0

Ignore the factor 6.
Subproblem 1
Set the factor '(1 + 24x + 24x2)' equal to zero and attempt to solve:

Simplifying
1 + 24x + 24x2 = 0

Solving
1 + 24x + 24x2 = 0

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

Divide each side by '24'.
0.04166666667 + x + x2 = 0

Move the constant term to the right:

Add '-0.04166666667' to each side of the equation.
0.04166666667 + x + -0.04166666667 + x2 = 0 + -0.04166666667

Reorder the terms:
0.04166666667 + -0.04166666667 + x + x2 = 0 + -0.04166666667

Combine like terms: 0.04166666667 + -0.04166666667 = 0.00000000000
0.00000000000 + x + x2 = 0 + -0.04166666667
x + x2 = 0 + -0.04166666667

Combine like terms: 0 + -0.04166666667 = -0.04166666667
x + x2 = -0.04166666667

The x term is x.  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.
 + 0.25 + x2 = -0.04166666667 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = -0.04166666667 + 0.25

Combine like terms: -0.04166666667 + 0.25 = 0.20833333333
1.25 + x2 = 0.20833333333

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.20833333333

Calculate the square root of the right side: 0.456435465

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.456435465 and -0.456435465.

Subproblem 1
x + 0.5 = 0.456435465

Simplifying
x + 0.5 = 0.456435465

Reorder the terms:
0.5 + x = 0.456435465

Solving
0.5 + x = 0.456435465

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 0.456435465 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.456435465 + -0.5
x = 0.456435465 + -0.5

Combine like terms: 0.456435465 + -0.5 = -0.043564535
x = -0.043564535

Simplifying
x = -0.043564535

Subproblem 2
x + 0.5 = -0.456435465

Simplifying
x + 0.5 = -0.456435465

Reorder the terms:
0.5 + x = -0.456435465

Solving
0.5 + x = -0.456435465

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -0.456435465 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.456435465 + -0.5
x = -0.456435465 + -0.5

Combine like terms: -0.456435465 + -0.5 = -0.956435465
x = -0.956435465

Simplifying
x = -0.956435465

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.043564535, -0.956435465}
Solution
x = {-0.043564535, -0.956435465}