3x^2-13x-6=0

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Solution for 3x^2-13x-6=0 equation: 


Simplifying
3x2 + -13x + -6 = 0

Reorder the terms:
-6 + -13x + 3x2 = 0

Solving
-6 + -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'.
-2 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -4.333333333x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -4.333333333x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -4.333333333x + x2 = 0 + 2
-4.333333333x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-4.333333333x + x2 = 2

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 = 2 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = 2 + 4.694444446

Combine like terms: 2 + 4.694444446 = 6.694444446
4.694444446 + -4.333333333x + x2 = 6.694444446

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 6.694444446

Calculate the square root of the right side: 2.58736245

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 2.58736245 and -2.58736245.


Subproblem 1
x + -2.166666667 = 2.58736245

Simplifying
x + -2.166666667 = 2.58736245

Reorder the terms:
-2.166666667 + x = 2.58736245

Solving
-2.166666667 + x = 2.58736245

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 = 2.58736245 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = 2.58736245 + 2.166666667
x = 2.58736245 + 2.166666667

Combine like terms: 2.58736245 + 2.166666667 = 4.754029117
x = 4.754029117

Simplifying
x = 4.754029117


Subproblem 2
x + -2.166666667 = -2.58736245

Simplifying
x + -2.166666667 = -2.58736245

Reorder the terms:
-2.166666667 + x = -2.58736245

Solving
-2.166666667 + x = -2.58736245

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 = -2.58736245 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = -2.58736245 + 2.166666667
x = -2.58736245 + 2.166666667

Combine like terms: -2.58736245 + 2.166666667 = -0.420695783
x = -0.420695783

Simplifying
x = -0.420695783


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.754029117, -0.420695783}

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