6x^2-10x-19=0

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Solution for 6x^2-10x-19=0 equation: 


Simplifying
6x2 + -10x + -19 = 0

Reorder the terms:
-19 + -10x + 6x2 = 0

Solving
-19 + -10x + 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 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '3.166666667' to each side of the equation.
-3.166666667 + -1.666666667x + 3.166666667 + x2 = 0 + 3.166666667

Reorder the terms:
-3.166666667 + 3.166666667 + -1.666666667x + x2 = 0 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + -1.666666667x + x2 = 0 + 3.166666667
-1.666666667x + x2 = 0 + 3.166666667

Combine like terms: 0 + 3.166666667 = 3.166666667
-1.666666667x + x2 = 3.166666667

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = 3.166666667 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 3.166666667 + 0.6944444447

Combine like terms: 3.166666667 + 0.6944444447 = 3.8611111117
0.6944444447 + -1.666666667x + x2 = 3.8611111117

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 3.8611111117

Calculate the square root of the right side: 1.964971021

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 1.964971021 and -1.964971021.


Subproblem 1
x + -0.8333333335 = 1.964971021

Simplifying
x + -0.8333333335 = 1.964971021

Reorder the terms:
-0.8333333335 + x = 1.964971021

Solving
-0.8333333335 + x = 1.964971021

Solving for variable 'x'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + x = 1.964971021 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = 1.964971021 + 0.8333333335
x = 1.964971021 + 0.8333333335

Combine like terms: 1.964971021 + 0.8333333335 = 2.7983043545
x = 2.7983043545

Simplifying
x = 2.7983043545


Subproblem 2
x + -0.8333333335 = -1.964971021

Simplifying
x + -0.8333333335 = -1.964971021

Reorder the terms:
-0.8333333335 + x = -1.964971021

Solving
-0.8333333335 + x = -1.964971021

Solving for variable 'x'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + x = -1.964971021 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = -1.964971021 + 0.8333333335
x = -1.964971021 + 0.8333333335

Combine like terms: -1.964971021 + 0.8333333335 = -1.1316376875
x = -1.1316376875

Simplifying
x = -1.1316376875


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.7983043545, -1.1316376875}

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