9x^2-114x-24=0

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Solution for 9x^2-114x-24=0 equation: 


Simplifying
9x2 + -114x + -24 = 0

Reorder the terms:
-24 + -114x + 9x2 = 0

Solving
-24 + -114x + 9x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-8 + -38x + 3x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-8 + -38x + 3x2)' equal to zero and attempt to solve:

Simplifying
-8 + -38x + 3x2 = 0

Solving
-8 + -38x + 3x2 = 0

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

Divide each side by '3'.
-2.666666667 + -12.66666667x + x2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + -12.66666667x + 2.666666667 + x2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + -12.66666667x + x2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + -12.66666667x + x2 = 0 + 2.666666667
-12.66666667x + x2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
-12.66666667x + x2 = 2.666666667

The x term is -12.66666667x.  Take half its coefficient (-6.333333335).
Square it (40.11111113) and add it to both sides.

Add '40.11111113' to each side of the equation.
-12.66666667x + 40.11111113 + x2 = 2.666666667 + 40.11111113

Reorder the terms:
40.11111113 + -12.66666667x + x2 = 2.666666667 + 40.11111113

Combine like terms: 2.666666667 + 40.11111113 = 42.777777797
40.11111113 + -12.66666667x + x2 = 42.777777797

Factor a perfect square on the left side:
(x + -6.333333335)(x + -6.333333335) = 42.777777797

Calculate the square root of the right side: 6.540472292

Break this problem into two subproblems by setting 
(x + -6.333333335) equal to 6.540472292 and -6.540472292.


Subproblem 1
x + -6.333333335 = 6.540472292

Simplifying
x + -6.333333335 = 6.540472292

Reorder the terms:
-6.333333335 + x = 6.540472292

Solving
-6.333333335 + x = 6.540472292

Solving for variable 'x'.

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

Add '6.333333335' to each side of the equation.
-6.333333335 + 6.333333335 + x = 6.540472292 + 6.333333335

Combine like terms: -6.333333335 + 6.333333335 = 0.000000000
0.000000000 + x = 6.540472292 + 6.333333335
x = 6.540472292 + 6.333333335

Combine like terms: 6.540472292 + 6.333333335 = 12.873805627
x = 12.873805627

Simplifying
x = 12.873805627


Subproblem 2
x + -6.333333335 = -6.540472292

Simplifying
x + -6.333333335 = -6.540472292

Reorder the terms:
-6.333333335 + x = -6.540472292

Solving
-6.333333335 + x = -6.540472292

Solving for variable 'x'.

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

Add '6.333333335' to each side of the equation.
-6.333333335 + 6.333333335 + x = -6.540472292 + 6.333333335

Combine like terms: -6.333333335 + 6.333333335 = 0.000000000
0.000000000 + x = -6.540472292 + 6.333333335
x = -6.540472292 + 6.333333335

Combine like terms: -6.540472292 + 6.333333335 = -0.207138957
x = -0.207138957

Simplifying
x = -0.207138957


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {12.873805627, -0.207138957}
Solution
x = {12.873805627, -0.207138957}

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