0=9x^2-39x+24

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Solution for 0=9x^2-39x+24 equation: 


Simplifying
0 = 9x2 + -39x + 24

Reorder the terms:
0 = 24 + -39x + 9x2

Solving
0 = 24 + -39x + 9x2

Solving for variable 'x'.

Combine like terms: 0 + -24 = -24
-24 + 39x + -9x2 = 24 + -39x + 9x2 + -24 + 39x + -9x2

Reorder the terms:
-24 + 39x + -9x2 = 24 + -24 + -39x + 39x + 9x2 + -9x2

Combine like terms: 24 + -24 = 0
-24 + 39x + -9x2 = 0 + -39x + 39x + 9x2 + -9x2
-24 + 39x + -9x2 = -39x + 39x + 9x2 + -9x2

Combine like terms: -39x + 39x = 0
-24 + 39x + -9x2 = 0 + 9x2 + -9x2
-24 + 39x + -9x2 = 9x2 + -9x2

Combine like terms: 9x2 + -9x2 = 0
-24 + 39x + -9x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-8 + 13x + -3x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-8 + 13x + -3x2)' equal to zero and attempt to solve:

Simplifying
-8 + 13x + -3x2 = 0

Solving
-8 + 13x + -3x2 = 0

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

Divide each side by '-3'.
2.666666667 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '-2.666666667' to each side of the equation.
2.666666667 + -4.333333333x + -2.666666667 + x2 = 0 + -2.666666667

Reorder the terms:
2.666666667 + -2.666666667 + -4.333333333x + x2 = 0 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + -4.333333333x + x2 = 0 + -2.666666667
-4.333333333x + x2 = 0 + -2.666666667

Combine like terms: 0 + -2.666666667 = -2.666666667
-4.333333333x + x2 = -2.666666667

The x term is -4.333333333x.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333x + 4.694444446 + x2 = -2.666666667 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = -2.666666667 + 4.694444446

Combine like terms: -2.666666667 + 4.694444446 = 2.027777779
4.694444446 + -4.333333333x + x2 = 2.027777779

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 2.027777779

Calculate the square root of the right side: 1.424000625

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 1.424000625 and -1.424000625.


Subproblem 1
x + -2.166666667 = 1.424000625

Simplifying
x + -2.166666667 = 1.424000625

Reorder the terms:
-2.166666667 + x = 1.424000625

Solving
-2.166666667 + x = 1.424000625

Solving for variable 'x'.

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

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = 1.424000625 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = 1.424000625 + 2.166666667
x = 1.424000625 + 2.166666667

Combine like terms: 1.424000625 + 2.166666667 = 3.590667292
x = 3.590667292

Simplifying
x = 3.590667292


Subproblem 2
x + -2.166666667 = -1.424000625

Simplifying
x + -2.166666667 = -1.424000625

Reorder the terms:
-2.166666667 + x = -1.424000625

Solving
-2.166666667 + x = -1.424000625

Solving for variable 'x'.

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

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = -1.424000625 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = -1.424000625 + 2.166666667
x = -1.424000625 + 2.166666667

Combine like terms: -1.424000625 + 2.166666667 = 0.742666042
x = 0.742666042

Simplifying
x = 0.742666042


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.590667292, 0.742666042}
Solution
x = {3.590667292, 0.742666042}

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