6x^2-17x-19=0

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Solution for 6x^2-17x-19=0 equation: 


Simplifying
6x2 + -17x + -19 = 0

Reorder the terms:
-19 + -17x + 6x2 = 0

Solving
-19 + -17x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-3.166666667 + -2.833333333x + x2 = 0

Move the constant term to the right:

Add '3.166666667' to each side of the equation.
-3.166666667 + -2.833333333x + 3.166666667 + x2 = 0 + 3.166666667

Reorder the terms:
-3.166666667 + 3.166666667 + -2.833333333x + x2 = 0 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + -2.833333333x + x2 = 0 + 3.166666667
-2.833333333x + x2 = 0 + 3.166666667

Combine like terms: 0 + 3.166666667 = 3.166666667
-2.833333333x + x2 = 3.166666667

The x term is -2.833333333x.  Take half its coefficient (-1.416666667).
Square it (2.006944445) and add it to both sides.

Add '2.006944445' to each side of the equation.
-2.833333333x + 2.006944445 + x2 = 3.166666667 + 2.006944445

Reorder the terms:
2.006944445 + -2.833333333x + x2 = 3.166666667 + 2.006944445

Combine like terms: 3.166666667 + 2.006944445 = 5.173611112
2.006944445 + -2.833333333x + x2 = 5.173611112

Factor a perfect square on the left side:
(x + -1.416666667)(x + -1.416666667) = 5.173611112

Calculate the square root of the right side: 2.274557344

Break this problem into two subproblems by setting 
(x + -1.416666667) equal to 2.274557344 and -2.274557344.


Subproblem 1
x + -1.416666667 = 2.274557344

Simplifying
x + -1.416666667 = 2.274557344

Reorder the terms:
-1.416666667 + x = 2.274557344

Solving
-1.416666667 + x = 2.274557344

Solving for variable 'x'.

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

Add '1.416666667' to each side of the equation.
-1.416666667 + 1.416666667 + x = 2.274557344 + 1.416666667

Combine like terms: -1.416666667 + 1.416666667 = 0.000000000
0.000000000 + x = 2.274557344 + 1.416666667
x = 2.274557344 + 1.416666667

Combine like terms: 2.274557344 + 1.416666667 = 3.691224011
x = 3.691224011

Simplifying
x = 3.691224011


Subproblem 2
x + -1.416666667 = -2.274557344

Simplifying
x + -1.416666667 = -2.274557344

Reorder the terms:
-1.416666667 + x = -2.274557344

Solving
-1.416666667 + x = -2.274557344

Solving for variable 'x'.

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

Add '1.416666667' to each side of the equation.
-1.416666667 + 1.416666667 + x = -2.274557344 + 1.416666667

Combine like terms: -1.416666667 + 1.416666667 = 0.000000000
0.000000000 + x = -2.274557344 + 1.416666667
x = -2.274557344 + 1.416666667

Combine like terms: -2.274557344 + 1.416666667 = -0.857890677
x = -0.857890677

Simplifying
x = -0.857890677


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.691224011, -0.857890677}

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