6n^2-20=n

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Solution for 6n^2-20=n equation: 


Simplifying
6n2 + -20 = n

Reorder the terms:
-20 + 6n2 = n

Solving
-20 + 6n2 = n

Solving for variable 'n'.

Reorder the terms:
-20 + -1n + 6n2 = n + -1n

Combine like terms: n + -1n = 0
-20 + -1n + 6n2 = 0

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

Divide each side by '6'.
-3.333333333 + -0.1666666667n + n2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + -0.1666666667n + 3.333333333 + n2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + -0.1666666667n + n2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + -0.1666666667n + n2 = 0 + 3.333333333
-0.1666666667n + n2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
-0.1666666667n + n2 = 3.333333333

The n term is -0.1666666667n.  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.
-0.1666666667n + 0.006944444447 + n2 = 3.333333333 + 0.006944444447

Reorder the terms:
0.006944444447 + -0.1666666667n + n2 = 3.333333333 + 0.006944444447

Combine like terms: 3.333333333 + 0.006944444447 = 3.340277777447
0.006944444447 + -0.1666666667n + n2 = 3.340277777447

Factor a perfect square on the left side:
(n + -0.08333333335)(n + -0.08333333335) = 3.340277777447

Calculate the square root of the right side: 1.827642683

Break this problem into two subproblems by setting 
(n + -0.08333333335) equal to 1.827642683 and -1.827642683.


Subproblem 1
n + -0.08333333335 = 1.827642683

Simplifying
n + -0.08333333335 = 1.827642683

Reorder the terms:
-0.08333333335 + n = 1.827642683

Solving
-0.08333333335 + n = 1.827642683

Solving for variable 'n'.

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

Add '0.08333333335' to each side of the equation.
-0.08333333335 + 0.08333333335 + n = 1.827642683 + 0.08333333335

Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000
0.00000000000 + n = 1.827642683 + 0.08333333335
n = 1.827642683 + 0.08333333335

Combine like terms: 1.827642683 + 0.08333333335 = 1.91097601635
n = 1.91097601635

Simplifying
n = 1.91097601635


Subproblem 2
n + -0.08333333335 = -1.827642683

Simplifying
n + -0.08333333335 = -1.827642683

Reorder the terms:
-0.08333333335 + n = -1.827642683

Solving
-0.08333333335 + n = -1.827642683

Solving for variable 'n'.

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

Add '0.08333333335' to each side of the equation.
-0.08333333335 + 0.08333333335 + n = -1.827642683 + 0.08333333335

Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000
0.00000000000 + n = -1.827642683 + 0.08333333335
n = -1.827642683 + 0.08333333335

Combine like terms: -1.827642683 + 0.08333333335 = -1.74430934965
n = -1.74430934965

Simplifying
n = -1.74430934965


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.91097601635, -1.74430934965}

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