9x^2+2x-6=0

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


Simplifying
9x2 + 2x + -6 = 0

Reorder the terms:
-6 + 2x + 9x2 = 0

Solving
-6 + 2x + 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'.
-0.6666666667 + 0.2222222222x + x2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + 0.2222222222x + 0.6666666667 + x2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + 0.2222222222x + x2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + 0.2222222222x + x2 = 0 + 0.6666666667
0.2222222222x + x2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
0.2222222222x + x2 = 0.6666666667

The x term is 0.2222222222x.  Take half its coefficient (0.1111111111).
Square it (0.01234567901) and add it to both sides.

Add '0.01234567901' to each side of the equation.
0.2222222222x + 0.01234567901 + x2 = 0.6666666667 + 0.01234567901

Reorder the terms:
0.01234567901 + 0.2222222222x + x2 = 0.6666666667 + 0.01234567901

Combine like terms: 0.6666666667 + 0.01234567901 = 0.67901234571
0.01234567901 + 0.2222222222x + x2 = 0.67901234571

Factor a perfect square on the left side:
(x + 0.1111111111)(x + 0.1111111111) = 0.67901234571

Calculate the square root of the right side: 0.824022054

Break this problem into two subproblems by setting 
(x + 0.1111111111) equal to 0.824022054 and -0.824022054.


Subproblem 1
x + 0.1111111111 = 0.824022054

Simplifying
x + 0.1111111111 = 0.824022054

Reorder the terms:
0.1111111111 + x = 0.824022054

Solving
0.1111111111 + x = 0.824022054

Solving for variable 'x'.

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

Add '-0.1111111111' to each side of the equation.
0.1111111111 + -0.1111111111 + x = 0.824022054 + -0.1111111111

Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000
0.0000000000 + x = 0.824022054 + -0.1111111111
x = 0.824022054 + -0.1111111111

Combine like terms: 0.824022054 + -0.1111111111 = 0.7129109429
x = 0.7129109429

Simplifying
x = 0.7129109429


Subproblem 2
x + 0.1111111111 = -0.824022054

Simplifying
x + 0.1111111111 = -0.824022054

Reorder the terms:
0.1111111111 + x = -0.824022054

Solving
0.1111111111 + x = -0.824022054

Solving for variable 'x'.

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

Add '-0.1111111111' to each side of the equation.
0.1111111111 + -0.1111111111 + x = -0.824022054 + -0.1111111111

Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000
0.0000000000 + x = -0.824022054 + -0.1111111111
x = -0.824022054 + -0.1111111111

Combine like terms: -0.824022054 + -0.1111111111 = -0.9351331651
x = -0.9351331651

Simplifying
x = -0.9351331651


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.7129109429, -0.9351331651}

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