3x^2-13x+9=0

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Solution for 3x^2-13x+9=0 equation: 


Simplifying
3x2 + -13x + 9 = 0

Reorder the terms:
9 + -13x + 3x2 = 0

Solving
9 + -13x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
3 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -4.333333333x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + -4.333333333x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -4.333333333x + x2 = 0 + -3
-4.333333333x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-4.333333333x + x2 = -3

The x term is -4.333333333x.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333x + 4.694444446 + x2 = -3 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = -3 + 4.694444446

Combine like terms: -3 + 4.694444446 = 1.694444446
4.694444446 + -4.333333333x + x2 = 1.694444446

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 1.694444446

Calculate the square root of the right side: 1.30170828

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 1.30170828 and -1.30170828.


Subproblem 1
x + -2.166666667 = 1.30170828

Simplifying
x + -2.166666667 = 1.30170828

Reorder the terms:
-2.166666667 + x = 1.30170828

Solving
-2.166666667 + x = 1.30170828

Solving for variable 'x'.

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

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = 1.30170828 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = 1.30170828 + 2.166666667
x = 1.30170828 + 2.166666667

Combine like terms: 1.30170828 + 2.166666667 = 3.468374947
x = 3.468374947

Simplifying
x = 3.468374947


Subproblem 2
x + -2.166666667 = -1.30170828

Simplifying
x + -2.166666667 = -1.30170828

Reorder the terms:
-2.166666667 + x = -1.30170828

Solving
-2.166666667 + x = -1.30170828

Solving for variable 'x'.

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

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = -1.30170828 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = -1.30170828 + 2.166666667
x = -1.30170828 + 2.166666667

Combine like terms: -1.30170828 + 2.166666667 = 0.864958387
x = 0.864958387

Simplifying
x = 0.864958387


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.468374947, 0.864958387}

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