24x^2-56x-32=0

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Solution for 24x^2-56x-32=0 equation: 


Simplifying
24x2 + -56x + -32 = 0

Reorder the terms:
-32 + -56x + 24x2 = 0

Solving
-32 + -56x + 24x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '8'.
8(-4 + -7x + 3x2) = 0

Ignore the factor 8.

Subproblem 1
Set the factor '(-4 + -7x + 3x2)' equal to zero and attempt to solve:

Simplifying
-4 + -7x + 3x2 = 0

Solving
-4 + -7x + 3x2 = 0

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

Divide each side by '3'.
-1.333333333 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -2.333333333x + 1.333333333 + x2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -2.333333333x + x2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -2.333333333x + x2 = 0 + 1.333333333
-2.333333333x + x2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-2.333333333x + x2 = 1.333333333

The x term is -2.333333333x.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333x + 1.361111112 + x2 = 1.333333333 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = 1.333333333 + 1.361111112

Combine like terms: 1.333333333 + 1.361111112 = 2.694444445
1.361111112 + -2.333333333x + x2 = 2.694444445

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 2.694444445

Calculate the square root of the right side: 1.6414763

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 1.6414763 and -1.6414763.


Subproblem 1
x + -1.166666667 = 1.6414763

Simplifying
x + -1.166666667 = 1.6414763

Reorder the terms:
-1.166666667 + x = 1.6414763

Solving
-1.166666667 + x = 1.6414763

Solving for variable 'x'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = 1.6414763 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = 1.6414763 + 1.166666667
x = 1.6414763 + 1.166666667

Combine like terms: 1.6414763 + 1.166666667 = 2.808142967
x = 2.808142967

Simplifying
x = 2.808142967


Subproblem 2
x + -1.166666667 = -1.6414763

Simplifying
x + -1.166666667 = -1.6414763

Reorder the terms:
-1.166666667 + x = -1.6414763

Solving
-1.166666667 + x = -1.6414763

Solving for variable 'x'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = -1.6414763 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = -1.6414763 + 1.166666667
x = -1.6414763 + 1.166666667

Combine like terms: -1.6414763 + 1.166666667 = -0.474809633
x = -0.474809633

Simplifying
x = -0.474809633


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.808142967, -0.474809633}
Solution
x = {2.808142967, -0.474809633}

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