24x^2-36x-32=0

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Solution for 24x^2-36x-32=0 equation: 


Simplifying
24x2 + -36x + -32 = 0

Reorder the terms:
-32 + -36x + 24x2 = 0

Solving
-32 + -36x + 24x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-8 + -9x + 6x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-8 + -9x + 6x2)' equal to zero and attempt to solve:

Simplifying
-8 + -9x + 6x2 = 0

Solving
-8 + -9x + 6x2 = 0

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

Divide each side by '6'.
-1.333333333 + -1.5x + x2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -1.5x + 1.333333333 + x2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -1.5x + x2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -1.5x + x2 = 0 + 1.333333333
-1.5x + x2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-1.5x + x2 = 1.333333333

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 1.333333333 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 1.333333333 + 0.5625

Combine like terms: 1.333333333 + 0.5625 = 1.895833333
0.5625 + -1.5x + x2 = 1.895833333

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 1.895833333

Calculate the square root of the right side: 1.376892637

Break this problem into two subproblems by setting 
(x + -0.75) equal to 1.376892637 and -1.376892637.


Subproblem 1
x + -0.75 = 1.376892637

Simplifying
x + -0.75 = 1.376892637

Reorder the terms:
-0.75 + x = 1.376892637

Solving
-0.75 + x = 1.376892637

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = 1.376892637 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 1.376892637 + 0.75
x = 1.376892637 + 0.75

Combine like terms: 1.376892637 + 0.75 = 2.126892637
x = 2.126892637

Simplifying
x = 2.126892637


Subproblem 2
x + -0.75 = -1.376892637

Simplifying
x + -0.75 = -1.376892637

Reorder the terms:
-0.75 + x = -1.376892637

Solving
-0.75 + x = -1.376892637

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = -1.376892637 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -1.376892637 + 0.75
x = -1.376892637 + 0.75

Combine like terms: -1.376892637 + 0.75 = -0.626892637
x = -0.626892637

Simplifying
x = -0.626892637


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.126892637, -0.626892637}
Solution
x = {2.126892637, -0.626892637}

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