-9x^2+6x+1=0

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Solution for -9x^2+6x+1=0 equation: 


Simplifying
-9x2 + 6x + 1 = 0

Reorder the terms:
1 + 6x + -9x2 = 0

Solving
1 + 6x + -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 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -0.6666666667x + 0.1111111111 + x2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -0.6666666667x + x2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -0.6666666667x + x2 = 0 + 0.1111111111
-0.6666666667x + x2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-0.6666666667x + x2 = 0.1111111111

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 = 0.1111111111 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 0.1111111111 + 0.1111111112

Combine like terms: 0.1111111111 + 0.1111111112 = 0.2222222223
0.1111111112 + -0.6666666667x + x2 = 0.2222222223

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 0.2222222223

Calculate the square root of the right side: 0.471404521

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 0.471404521 and -0.471404521.


Subproblem 1
x + -0.3333333334 = 0.471404521

Simplifying
x + -0.3333333334 = 0.471404521

Reorder the terms:
-0.3333333334 + x = 0.471404521

Solving
-0.3333333334 + x = 0.471404521

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 = 0.471404521 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = 0.471404521 + 0.3333333334
x = 0.471404521 + 0.3333333334

Combine like terms: 0.471404521 + 0.3333333334 = 0.8047378544
x = 0.8047378544

Simplifying
x = 0.8047378544


Subproblem 2
x + -0.3333333334 = -0.471404521

Simplifying
x + -0.3333333334 = -0.471404521

Reorder the terms:
-0.3333333334 + x = -0.471404521

Solving
-0.3333333334 + x = -0.471404521

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 = -0.471404521 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = -0.471404521 + 0.3333333334
x = -0.471404521 + 0.3333333334

Combine like terms: -0.471404521 + 0.3333333334 = -0.1380711876
x = -0.1380711876

Simplifying
x = -0.1380711876


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.8047378544, -0.1380711876}

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