32x^2+24x-4=0

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Solution for 32x^2+24x-4=0 equation: 


Simplifying
32x2 + 24x + -4 = 0

Reorder the terms:
-4 + 24x + 32x2 = 0

Solving
-4 + 24x + 32x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-1 + 6x + 8x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-1 + 6x + 8x2)' equal to zero and attempt to solve:

Simplifying
-1 + 6x + 8x2 = 0

Solving
-1 + 6x + 8x2 = 0

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

Divide each side by '8'.
-0.125 + 0.75x + x2 = 0

Move the constant term to the right:

Add '0.125' to each side of the equation.
-0.125 + 0.75x + 0.125 + x2 = 0 + 0.125

Reorder the terms:
-0.125 + 0.125 + 0.75x + x2 = 0 + 0.125

Combine like terms: -0.125 + 0.125 = 0.000
0.000 + 0.75x + x2 = 0 + 0.125
0.75x + x2 = 0 + 0.125

Combine like terms: 0 + 0.125 = 0.125
0.75x + x2 = 0.125

The x term is 0.75x.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75x + 0.140625 + x2 = 0.125 + 0.140625

Reorder the terms:
0.140625 + 0.75x + x2 = 0.125 + 0.140625

Combine like terms: 0.125 + 0.140625 = 0.265625
0.140625 + 0.75x + x2 = 0.265625

Factor a perfect square on the left side:
(x + 0.375)(x + 0.375) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(x + 0.375) equal to 0.515388203 and -0.515388203.


Subproblem 1
x + 0.375 = 0.515388203

Simplifying
x + 0.375 = 0.515388203

Reorder the terms:
0.375 + x = 0.515388203

Solving
0.375 + x = 0.515388203

Solving for variable 'x'.

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

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = 0.515388203 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = 0.515388203 + -0.375
x = 0.515388203 + -0.375

Combine like terms: 0.515388203 + -0.375 = 0.140388203
x = 0.140388203

Simplifying
x = 0.140388203


Subproblem 2
x + 0.375 = -0.515388203

Simplifying
x + 0.375 = -0.515388203

Reorder the terms:
0.375 + x = -0.515388203

Solving
0.375 + x = -0.515388203

Solving for variable 'x'.

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

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = -0.515388203 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = -0.515388203 + -0.375
x = -0.515388203 + -0.375

Combine like terms: -0.515388203 + -0.375 = -0.890388203
x = -0.890388203

Simplifying
x = -0.890388203


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.140388203, -0.890388203}
Solution
x = {0.140388203, -0.890388203}

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