9x^2-2x-3=0

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Solution for 9x^2-2x-3=0 equation: 


Simplifying
9x2 + -2x + -3 = 0

Reorder the terms:
-3 + -2x + 9x2 = 0

Solving
-3 + -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'.
-0.3333333333 + -0.2222222222x + x2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -0.2222222222x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -0.2222222222x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -0.2222222222x + x2 = 0 + 0.3333333333
-0.2222222222x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-0.2222222222x + x2 = 0.3333333333

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 = 0.3333333333 + 0.01234567901

Reorder the terms:
0.01234567901 + -0.2222222222x + x2 = 0.3333333333 + 0.01234567901

Combine like terms: 0.3333333333 + 0.01234567901 = 0.34567901231
0.01234567901 + -0.2222222222x + x2 = 0.34567901231

Factor a perfect square on the left side:
(x + -0.1111111111)(x + -0.1111111111) = 0.34567901231

Calculate the square root of the right side: 0.587944736

Break this problem into two subproblems by setting 
(x + -0.1111111111) equal to 0.587944736 and -0.587944736.


Subproblem 1
x + -0.1111111111 = 0.587944736

Simplifying
x + -0.1111111111 = 0.587944736

Reorder the terms:
-0.1111111111 + x = 0.587944736

Solving
-0.1111111111 + x = 0.587944736

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 = 0.587944736 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + x = 0.587944736 + 0.1111111111
x = 0.587944736 + 0.1111111111

Combine like terms: 0.587944736 + 0.1111111111 = 0.6990558471
x = 0.6990558471

Simplifying
x = 0.6990558471


Subproblem 2
x + -0.1111111111 = -0.587944736

Simplifying
x + -0.1111111111 = -0.587944736

Reorder the terms:
-0.1111111111 + x = -0.587944736

Solving
-0.1111111111 + x = -0.587944736

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 = -0.587944736 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + x = -0.587944736 + 0.1111111111
x = -0.587944736 + 0.1111111111

Combine like terms: -0.587944736 + 0.1111111111 = -0.4768336249
x = -0.4768336249

Simplifying
x = -0.4768336249


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.6990558471, -0.4768336249}

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