(9x^2)+3x-1=0

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Solution for (9x^2)+3x-1=0 equation: 


Simplifying
(9x2) + 3x + -1 = 0

Reorder the terms:
-1 + 3x + (9x2) = 0

Solving
-1 + 3x + (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'.
-0.1111111111 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + 0.3333333333x + 0.1111111111 + x2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + 0.3333333333x + x2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + 0.3333333333x + x2 = 0 + 0.1111111111
0.3333333333x + x2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
0.3333333333x + x2 = 0.1111111111

The x term is 0.3333333333x.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333x + 0.02777777779 + x2 = 0.1111111111 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333x + x2 = 0.1111111111 + 0.02777777779

Combine like terms: 0.1111111111 + 0.02777777779 = 0.13888888889
0.02777777779 + 0.3333333333x + x2 = 0.13888888889

Factor a perfect square on the left side:
((x) + 0.1666666667)((x) + 0.1666666667) = 0.13888888889

Calculate the square root of the right side: 0.372677996

Break this problem into two subproblems by setting 
((x) + 0.1666666667) equal to 0.372677996 and -0.372677996.


Subproblem 1
(x) + 0.1666666667 = 0.372677996

Simplifying
(x) + 0.1666666667 = 0.372677996
x + 0.1666666667 = 0.372677996

Reorder the terms:
0.1666666667 + x = 0.372677996

Solving
0.1666666667 + x = 0.372677996

Solving for variable 'x'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + x = 0.372677996 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + x = 0.372677996 + -0.1666666667
x = 0.372677996 + -0.1666666667

Combine like terms: 0.372677996 + -0.1666666667 = 0.2060113293
x = 0.2060113293

Simplifying
x = 0.2060113293


Subproblem 2
(x) + 0.1666666667 = -0.372677996

Simplifying
(x) + 0.1666666667 = -0.372677996
x + 0.1666666667 = -0.372677996

Reorder the terms:
0.1666666667 + x = -0.372677996

Solving
0.1666666667 + x = -0.372677996

Solving for variable 'x'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + x = -0.372677996 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + x = -0.372677996 + -0.1666666667
x = -0.372677996 + -0.1666666667

Combine like terms: -0.372677996 + -0.1666666667 = -0.5393446627
x = -0.5393446627

Simplifying
x = -0.5393446627


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.2060113293, -0.5393446627}

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