6X^2-14X-5=0

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Solution for 6X^2-14X-5=0 equation: 


Simplifying
6X2 + -14X + -5 = 0

Reorder the terms:
-5 + -14X + 6X2 = 0

Solving
-5 + -14X + 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'.
-0.8333333333 + -2.333333333X + X2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + -2.333333333X + 0.8333333333 + X2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + -2.333333333X + X2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + -2.333333333X + X2 = 0 + 0.8333333333
-2.333333333X + X2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
-2.333333333X + X2 = 0.8333333333

The X term is -2.333333333X.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333X + 1.361111112 + X2 = 0.8333333333 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333X + X2 = 0.8333333333 + 1.361111112

Combine like terms: 0.8333333333 + 1.361111112 = 2.1944444453
1.361111112 + -2.333333333X + X2 = 2.1944444453

Factor a perfect square on the left side:
(X + -1.166666667)(X + -1.166666667) = 2.1944444453

Calculate the square root of the right side: 1.481365737

Break this problem into two subproblems by setting 
(X + -1.166666667) equal to 1.481365737 and -1.481365737.


Subproblem 1
X + -1.166666667 = 1.481365737

Simplifying
X + -1.166666667 = 1.481365737

Reorder the terms:
-1.166666667 + X = 1.481365737

Solving
-1.166666667 + X = 1.481365737

Solving for variable 'X'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + X = 1.481365737 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + X = 1.481365737 + 1.166666667
X = 1.481365737 + 1.166666667

Combine like terms: 1.481365737 + 1.166666667 = 2.648032404
X = 2.648032404

Simplifying
X = 2.648032404


Subproblem 2
X + -1.166666667 = -1.481365737

Simplifying
X + -1.166666667 = -1.481365737

Reorder the terms:
-1.166666667 + X = -1.481365737

Solving
-1.166666667 + X = -1.481365737

Solving for variable 'X'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + X = -1.481365737 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + X = -1.481365737 + 1.166666667
X = -1.481365737 + 1.166666667

Combine like terms: -1.481365737 + 1.166666667 = -0.31469907
X = -0.31469907

Simplifying
X = -0.31469907


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {2.648032404, -0.31469907}

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