9x^2-6x-75=0

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Solution for 9x^2-6x-75=0 equation: 


Simplifying
9x2 + -6x + -75 = 0

Reorder the terms:
-75 + -6x + 9x2 = 0

Solving
-75 + -6x + 9x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-25 + -2x + 3x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-25 + -2x + 3x2)' equal to zero and attempt to solve:

Simplifying
-25 + -2x + 3x2 = 0

Solving
-25 + -2x + 3x2 = 0

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

Divide each side by '3'.
-8.333333333 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '8.333333333' to each side of the equation.
-8.333333333 + -0.6666666667x + 8.333333333 + x2 = 0 + 8.333333333

Reorder the terms:
-8.333333333 + 8.333333333 + -0.6666666667x + x2 = 0 + 8.333333333

Combine like terms: -8.333333333 + 8.333333333 = 0.000000000
0.000000000 + -0.6666666667x + x2 = 0 + 8.333333333
-0.6666666667x + x2 = 0 + 8.333333333

Combine like terms: 0 + 8.333333333 = 8.333333333
-0.6666666667x + x2 = 8.333333333

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 8.333333333 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 8.333333333 + 0.1111111112

Combine like terms: 8.333333333 + 0.1111111112 = 8.4444444442
0.1111111112 + -0.6666666667x + x2 = 8.4444444442

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 8.4444444442

Calculate the square root of the right side: 2.905932629

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 2.905932629 and -2.905932629.


Subproblem 1
x + -0.3333333334 = 2.905932629

Simplifying
x + -0.3333333334 = 2.905932629

Reorder the terms:
-0.3333333334 + x = 2.905932629

Solving
-0.3333333334 + x = 2.905932629

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = 2.905932629 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = 2.905932629 + 0.3333333334
x = 2.905932629 + 0.3333333334

Combine like terms: 2.905932629 + 0.3333333334 = 3.2392659624
x = 3.2392659624

Simplifying
x = 3.2392659624


Subproblem 2
x + -0.3333333334 = -2.905932629

Simplifying
x + -0.3333333334 = -2.905932629

Reorder the terms:
-0.3333333334 + x = -2.905932629

Solving
-0.3333333334 + x = -2.905932629

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = -2.905932629 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = -2.905932629 + 0.3333333334
x = -2.905932629 + 0.3333333334

Combine like terms: -2.905932629 + 0.3333333334 = -2.5725992956
x = -2.5725992956

Simplifying
x = -2.5725992956


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.2392659624, -2.5725992956}
Solution
x = {3.2392659624, -2.5725992956}

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