24x^2-12x-92=0

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Solution for 24x^2-12x-92=0 equation: 


Simplifying
24x2 + -12x + -92 = 0

Reorder the terms:
-92 + -12x + 24x2 = 0

Solving
-92 + -12x + 24x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-23 + -3x + 6x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-23 + -3x + 6x2)' equal to zero and attempt to solve:

Simplifying
-23 + -3x + 6x2 = 0

Solving
-23 + -3x + 6x2 = 0

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

Divide each side by '6'.
-3.833333333 + -0.5x + x2 = 0

Move the constant term to the right:

Add '3.833333333' to each side of the equation.
-3.833333333 + -0.5x + 3.833333333 + x2 = 0 + 3.833333333

Reorder the terms:
-3.833333333 + 3.833333333 + -0.5x + x2 = 0 + 3.833333333

Combine like terms: -3.833333333 + 3.833333333 = 0.000000000
0.000000000 + -0.5x + x2 = 0 + 3.833333333
-0.5x + x2 = 0 + 3.833333333

Combine like terms: 0 + 3.833333333 = 3.833333333
-0.5x + x2 = 3.833333333

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 3.833333333 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 3.833333333 + 0.0625

Combine like terms: 3.833333333 + 0.0625 = 3.895833333
0.0625 + -0.5x + x2 = 3.895833333

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 3.895833333

Calculate the square root of the right side: 1.973786547

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.973786547 and -1.973786547.


Subproblem 1
x + -0.25 = 1.973786547

Simplifying
x + -0.25 = 1.973786547

Reorder the terms:
-0.25 + x = 1.973786547

Solving
-0.25 + x = 1.973786547

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = 1.973786547 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = 1.973786547 + 0.25
x = 1.973786547 + 0.25

Combine like terms: 1.973786547 + 0.25 = 2.223786547
x = 2.223786547

Simplifying
x = 2.223786547


Subproblem 2
x + -0.25 = -1.973786547

Simplifying
x + -0.25 = -1.973786547

Reorder the terms:
-0.25 + x = -1.973786547

Solving
-0.25 + x = -1.973786547

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = -1.973786547 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = -1.973786547 + 0.25
x = -1.973786547 + 0.25

Combine like terms: -1.973786547 + 0.25 = -1.723786547
x = -1.723786547

Simplifying
x = -1.723786547


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.223786547, -1.723786547}
Solution
x = {2.223786547, -1.723786547}

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