9x^2-61x-75=0

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Solution for 9x^2-61x-75=0 equation: 


Simplifying
9x2 + -61x + -75 = 0

Reorder the terms:
-75 + -61x + 9x2 = 0

Solving
-75 + -61x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
-8.333333333 + -6.777777778x + x2 = 0

Move the constant term to the right:

Add '8.333333333' to each side of the equation.
-8.333333333 + -6.777777778x + 8.333333333 + x2 = 0 + 8.333333333

Reorder the terms:
-8.333333333 + 8.333333333 + -6.777777778x + x2 = 0 + 8.333333333

Combine like terms: -8.333333333 + 8.333333333 = 0.000000000
0.000000000 + -6.777777778x + x2 = 0 + 8.333333333
-6.777777778x + x2 = 0 + 8.333333333

Combine like terms: 0 + 8.333333333 = 8.333333333
-6.777777778x + x2 = 8.333333333

The x term is -6.777777778x.  Take half its coefficient (-3.388888889).
Square it (11.48456790) and add it to both sides.

Add '11.48456790' to each side of the equation.
-6.777777778x + 11.48456790 + x2 = 8.333333333 + 11.48456790

Reorder the terms:
11.48456790 + -6.777777778x + x2 = 8.333333333 + 11.48456790

Combine like terms: 8.333333333 + 11.48456790 = 19.817901233
11.48456790 + -6.777777778x + x2 = 19.817901233

Factor a perfect square on the left side:
(x + -3.388888889)(x + -3.388888889) = 19.817901233

Calculate the square root of the right side: 4.451730139

Break this problem into two subproblems by setting 
(x + -3.388888889) equal to 4.451730139 and -4.451730139.


Subproblem 1
x + -3.388888889 = 4.451730139

Simplifying
x + -3.388888889 = 4.451730139

Reorder the terms:
-3.388888889 + x = 4.451730139

Solving
-3.388888889 + x = 4.451730139

Solving for variable 'x'.

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

Add '3.388888889' to each side of the equation.
-3.388888889 + 3.388888889 + x = 4.451730139 + 3.388888889

Combine like terms: -3.388888889 + 3.388888889 = 0.000000000
0.000000000 + x = 4.451730139 + 3.388888889
x = 4.451730139 + 3.388888889

Combine like terms: 4.451730139 + 3.388888889 = 7.840619028
x = 7.840619028

Simplifying
x = 7.840619028


Subproblem 2
x + -3.388888889 = -4.451730139

Simplifying
x + -3.388888889 = -4.451730139

Reorder the terms:
-3.388888889 + x = -4.451730139

Solving
-3.388888889 + x = -4.451730139

Solving for variable 'x'.

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

Add '3.388888889' to each side of the equation.
-3.388888889 + 3.388888889 + x = -4.451730139 + 3.388888889

Combine like terms: -3.388888889 + 3.388888889 = 0.000000000
0.000000000 + x = -4.451730139 + 3.388888889
x = -4.451730139 + 3.388888889

Combine like terms: -4.451730139 + 3.388888889 = -1.06284125
x = -1.06284125

Simplifying
x = -1.06284125


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.840619028, -1.06284125}

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