24x^2-6x-12=0

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Solution for 24x^2-6x-12=0 equation: 


Simplifying
24x2 + -6x + -12 = 0

Reorder the terms:
-12 + -6x + 24x2 = 0

Solving
-12 + -6x + 24x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6'.
6(-2 + -1x + 4x2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(-2 + -1x + 4x2)' equal to zero and attempt to solve:

Simplifying
-2 + -1x + 4x2 = 0

Solving
-2 + -1x + 4x2 = 0

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

Divide each side by '4'.
-0.5 + -0.25x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.25x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.25x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.25x + x2 = 0 + 0.5
-0.25x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.25x + x2 = 0.5

The x term is -0.25x.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25x + 0.015625 + x2 = 0.5 + 0.015625

Reorder the terms:
0.015625 + -0.25x + x2 = 0.5 + 0.015625

Combine like terms: 0.5 + 0.015625 = 0.515625
0.015625 + -0.25x + x2 = 0.515625

Factor a perfect square on the left side:
(x + -0.125)(x + -0.125) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(x + -0.125) equal to 0.718070331 and -0.718070331.


Subproblem 1
x + -0.125 = 0.718070331

Simplifying
x + -0.125 = 0.718070331

Reorder the terms:
-0.125 + x = 0.718070331

Solving
-0.125 + x = 0.718070331

Solving for variable 'x'.

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

Add '0.125' to each side of the equation.
-0.125 + 0.125 + x = 0.718070331 + 0.125

Combine like terms: -0.125 + 0.125 = 0.000
0.000 + x = 0.718070331 + 0.125
x = 0.718070331 + 0.125

Combine like terms: 0.718070331 + 0.125 = 0.843070331
x = 0.843070331

Simplifying
x = 0.843070331


Subproblem 2
x + -0.125 = -0.718070331

Simplifying
x + -0.125 = -0.718070331

Reorder the terms:
-0.125 + x = -0.718070331

Solving
-0.125 + x = -0.718070331

Solving for variable 'x'.

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

Add '0.125' to each side of the equation.
-0.125 + 0.125 + x = -0.718070331 + 0.125

Combine like terms: -0.125 + 0.125 = 0.000
0.000 + x = -0.718070331 + 0.125
x = -0.718070331 + 0.125

Combine like terms: -0.718070331 + 0.125 = -0.593070331
x = -0.593070331

Simplifying
x = -0.593070331


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.843070331, -0.593070331}
Solution
x = {0.843070331, -0.593070331}

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