3x^2-19x-6=0

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Solution for 3x^2-19x-6=0 equation: 


Simplifying
3x2 + -19x + -6 = 0

Reorder the terms:
-6 + -19x + 3x2 = 0

Solving
-6 + -19x + 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 + -6.333333333x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -6.333333333x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -6.333333333x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -6.333333333x + x2 = 0 + 2
-6.333333333x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-6.333333333x + x2 = 2

The x term is -6.333333333x.  Take half its coefficient (-3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
-6.333333333x + 10.02777778 + x2 = 2 + 10.02777778

Reorder the terms:
10.02777778 + -6.333333333x + x2 = 2 + 10.02777778

Combine like terms: 2 + 10.02777778 = 12.02777778
10.02777778 + -6.333333333x + x2 = 12.02777778

Factor a perfect square on the left side:
(x + -3.166666667)(x + -3.166666667) = 12.02777778

Calculate the square root of the right side: 3.468108675

Break this problem into two subproblems by setting 
(x + -3.166666667) equal to 3.468108675 and -3.468108675.


Subproblem 1
x + -3.166666667 = 3.468108675

Simplifying
x + -3.166666667 = 3.468108675

Reorder the terms:
-3.166666667 + x = 3.468108675

Solving
-3.166666667 + x = 3.468108675

Solving for variable 'x'.

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

Add '3.166666667' to each side of the equation.
-3.166666667 + 3.166666667 + x = 3.468108675 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + x = 3.468108675 + 3.166666667
x = 3.468108675 + 3.166666667

Combine like terms: 3.468108675 + 3.166666667 = 6.634775342
x = 6.634775342

Simplifying
x = 6.634775342


Subproblem 2
x + -3.166666667 = -3.468108675

Simplifying
x + -3.166666667 = -3.468108675

Reorder the terms:
-3.166666667 + x = -3.468108675

Solving
-3.166666667 + x = -3.468108675

Solving for variable 'x'.

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

Add '3.166666667' to each side of the equation.
-3.166666667 + 3.166666667 + x = -3.468108675 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + x = -3.468108675 + 3.166666667
x = -3.468108675 + 3.166666667

Combine like terms: -3.468108675 + 3.166666667 = -0.301442008
x = -0.301442008

Simplifying
x = -0.301442008


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.634775342, -0.301442008}

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