6x^2-x-37=0

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Solution for 6x^2-x-37=0 equation: 


Simplifying
6x2 + -1x + -37 = 0

Reorder the terms:
-37 + -1x + 6x2 = 0

Solving
-37 + -1x + 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'.
-6.166666667 + -0.1666666667x + x2 = 0

Move the constant term to the right:

Add '6.166666667' to each side of the equation.
-6.166666667 + -0.1666666667x + 6.166666667 + x2 = 0 + 6.166666667

Reorder the terms:
-6.166666667 + 6.166666667 + -0.1666666667x + x2 = 0 + 6.166666667

Combine like terms: -6.166666667 + 6.166666667 = 0.000000000
0.000000000 + -0.1666666667x + x2 = 0 + 6.166666667
-0.1666666667x + x2 = 0 + 6.166666667

Combine like terms: 0 + 6.166666667 = 6.166666667
-0.1666666667x + x2 = 6.166666667

The x term is -0.1666666667x.  Take half its coefficient (-0.08333333335).
Square it (0.006944444447) and add it to both sides.

Add '0.006944444447' to each side of the equation.
-0.1666666667x + 0.006944444447 + x2 = 6.166666667 + 0.006944444447

Reorder the terms:
0.006944444447 + -0.1666666667x + x2 = 6.166666667 + 0.006944444447

Combine like terms: 6.166666667 + 0.006944444447 = 6.173611111447
0.006944444447 + -0.1666666667x + x2 = 6.173611111447

Factor a perfect square on the left side:
(x + -0.08333333335)(x + -0.08333333335) = 6.173611111447

Calculate the square root of the right side: 2.484675253

Break this problem into two subproblems by setting 
(x + -0.08333333335) equal to 2.484675253 and -2.484675253.


Subproblem 1
x + -0.08333333335 = 2.484675253

Simplifying
x + -0.08333333335 = 2.484675253

Reorder the terms:
-0.08333333335 + x = 2.484675253

Solving
-0.08333333335 + x = 2.484675253

Solving for variable 'x'.

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

Add '0.08333333335' to each side of the equation.
-0.08333333335 + 0.08333333335 + x = 2.484675253 + 0.08333333335

Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000
0.00000000000 + x = 2.484675253 + 0.08333333335
x = 2.484675253 + 0.08333333335

Combine like terms: 2.484675253 + 0.08333333335 = 2.56800858635
x = 2.56800858635

Simplifying
x = 2.56800858635


Subproblem 2
x + -0.08333333335 = -2.484675253

Simplifying
x + -0.08333333335 = -2.484675253

Reorder the terms:
-0.08333333335 + x = -2.484675253

Solving
-0.08333333335 + x = -2.484675253

Solving for variable 'x'.

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

Add '0.08333333335' to each side of the equation.
-0.08333333335 + 0.08333333335 + x = -2.484675253 + 0.08333333335

Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000
0.00000000000 + x = -2.484675253 + 0.08333333335
x = -2.484675253 + 0.08333333335

Combine like terms: -2.484675253 + 0.08333333335 = -2.40134191965
x = -2.40134191965

Simplifying
x = -2.40134191965


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.56800858635, -2.40134191965}

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