n^2+1=5n

Solution for n^2+1=5n equation:

Simplifying
n2 + 1 = 5n

Reorder the terms:
1 + n2 = 5n

Solving
1 + n2 = 5n

Solving for variable 'n'.

Reorder the terms:
1 + -5n + n2 = 5n + -5n

Combine like terms: 5n + -5n = 0
1 + -5n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -5n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + -5n + n2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -5n + n2 = 0 + -1
-5n + n2 = 0 + -1

Combine like terms: 0 + -1 = -1
-5n + n2 = -1

The n term is -5n.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5n + 6.25 + n2 = -1 + 6.25

Reorder the terms:
6.25 + -5n + n2 = -1 + 6.25

Combine like terms: -1 + 6.25 = 5.25
6.25 + -5n + n2 = 5.25

Factor a perfect square on the left side:
(n + -2.5)(n + -2.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(n + -2.5) equal to 2.291287847 and -2.291287847.

Subproblem 1
n + -2.5 = 2.291287847

Simplifying
n + -2.5 = 2.291287847

Reorder the terms:
-2.5 + n = 2.291287847

Solving
-2.5 + n = 2.291287847

Solving for variable 'n'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + n = 2.291287847 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + n = 2.291287847 + 2.5
n = 2.291287847 + 2.5

Combine like terms: 2.291287847 + 2.5 = 4.791287847
n = 4.791287847

Simplifying
n = 4.791287847

Subproblem 2
n + -2.5 = -2.291287847

Simplifying
n + -2.5 = -2.291287847

Reorder the terms:
-2.5 + n = -2.291287847

Solving
-2.5 + n = -2.291287847

Solving for variable 'n'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + n = -2.291287847 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + n = -2.291287847 + 2.5
n = -2.291287847 + 2.5

Combine like terms: -2.291287847 + 2.5 = 0.208712153
n = 0.208712153

Simplifying
n = 0.208712153

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.791287847, 0.208712153}