3n+1=n^2-27

Solution for 3n+1=n^2-27 equation:

Simplifying
3n + 1 = n2 + -27

Reorder the terms:
1 + 3n = n2 + -27

Reorder the terms:
1 + 3n = -27 + n2

Solving
1 + 3n = -27 + n2

Solving for variable 'n'.

Reorder the terms:
1 + 27 + 3n + -1n2 = -27 + n2 + 27 + -1n2

Combine like terms: 1 + 27 = 28
28 + 3n + -1n2 = -27 + n2 + 27 + -1n2

Reorder the terms:
28 + 3n + -1n2 = -27 + 27 + n2 + -1n2

Combine like terms: -27 + 27 = 0
28 + 3n + -1n2 = 0 + n2 + -1n2
28 + 3n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
28 + 3n + -1n2 = 0

Factor a trinomial.
(7 + -1n)(4 + n) = 0

Subproblem 1
Set the factor '(7 + -1n)' equal to zero and attempt to solve:

Simplifying
7 + -1n = 0

Solving
7 + -1n = 0

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

Add '-7' to each side of the equation.
7 + -7 + -1n = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -1n = 0 + -7
-1n = 0 + -7

Combine like terms: 0 + -7 = -7
-1n = -7

Divide each side by '-1'.
n = 7

Simplifying
n = 7
Subproblem 2
Set the factor '(4 + n)' equal to zero and attempt to solve:

Simplifying
4 + n = 0

Solving
4 + n = 0

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

Add '-4' to each side of the equation.
4 + -4 + n = 0 + -4

Combine like terms: 4 + -4 = 0
0 + n = 0 + -4
n = 0 + -4

Combine like terms: 0 + -4 = -4
n = -4

Simplifying
n = -4
Solution
n = {7, -4}