6x^2-5x-90=0

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Solution for 6x^2-5x-90=0 equation: 


Simplifying
6x2 + -5x + -90 = 0

Reorder the terms:
-90 + -5x + 6x2 = 0

Solving
-90 + -5x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-15 + -0.8333333333x + x2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + -0.8333333333x + 15 + x2 = 0 + 15

Reorder the terms:
-15 + 15 + -0.8333333333x + x2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + -0.8333333333x + x2 = 0 + 15
-0.8333333333x + x2 = 0 + 15

Combine like terms: 0 + 15 = 15
-0.8333333333x + x2 = 15

The x term is -0.8333333333x.  Take half its coefficient (-0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
-0.8333333333x + 0.1736111111 + x2 = 15 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333x + x2 = 15 + 0.1736111111

Combine like terms: 15 + 0.1736111111 = 15.1736111111
0.1736111111 + -0.8333333333x + x2 = 15.1736111111

Factor a perfect square on the left side:
(x + -0.4166666667)(x + -0.4166666667) = 15.1736111111

Calculate the square root of the right side: 3.895331964

Break this problem into two subproblems by setting 
(x + -0.4166666667) equal to 3.895331964 and -3.895331964.


Subproblem 1
x + -0.4166666667 = 3.895331964

Simplifying
x + -0.4166666667 = 3.895331964

Reorder the terms:
-0.4166666667 + x = 3.895331964

Solving
-0.4166666667 + x = 3.895331964

Solving for variable 'x'.

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

Add '0.4166666667' to each side of the equation.
-0.4166666667 + 0.4166666667 + x = 3.895331964 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + x = 3.895331964 + 0.4166666667
x = 3.895331964 + 0.4166666667

Combine like terms: 3.895331964 + 0.4166666667 = 4.3119986307
x = 4.3119986307

Simplifying
x = 4.3119986307


Subproblem 2
x + -0.4166666667 = -3.895331964

Simplifying
x + -0.4166666667 = -3.895331964

Reorder the terms:
-0.4166666667 + x = -3.895331964

Solving
-0.4166666667 + x = -3.895331964

Solving for variable 'x'.

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

Add '0.4166666667' to each side of the equation.
-0.4166666667 + 0.4166666667 + x = -3.895331964 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + x = -3.895331964 + 0.4166666667
x = -3.895331964 + 0.4166666667

Combine like terms: -3.895331964 + 0.4166666667 = -3.4786652973
x = -3.4786652973

Simplifying
x = -3.4786652973


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.3119986307, -3.4786652973}

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