6n(n+1)=3-n

Solution for 6n(n+1)=3-n equation:

Simplifying
6n(n + 1) = 3 + -1n

Reorder the terms:
6n(1 + n) = 3 + -1n
(1 * 6n + n * 6n) = 3 + -1n
(6n + 6n2) = 3 + -1n

Solving
6n + 6n2 = 3 + -1n

Solving for variable 'n'.

Reorder the terms:
-3 + 6n + n + 6n2 = 3 + -1n + -3 + n

Combine like terms: 6n + n = 7n
-3 + 7n + 6n2 = 3 + -1n + -3 + n

Reorder the terms:
-3 + 7n + 6n2 = 3 + -3 + -1n + n

Combine like terms: 3 + -3 = 0
-3 + 7n + 6n2 = 0 + -1n + n
-3 + 7n + 6n2 = -1n + n

Combine like terms: -1n + n = 0
-3 + 7n + 6n2 = 0

Factor a trinomial.
(-3 + -2n)(1 + -3n) = 0

Subproblem 1
Set the factor '(-3 + -2n)' equal to zero and attempt to solve:

Simplifying
-3 + -2n = 0

Solving
-3 + -2n = 0

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

Add '3' to each side of the equation.
-3 + 3 + -2n = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -2n = 0 + 3
-2n = 0 + 3

Combine like terms: 0 + 3 = 3
-2n = 3

Divide each side by '-2'.
n = -1.5

Simplifying
n = -1.5
Subproblem 2
Set the factor '(1 + -3n)' equal to zero and attempt to solve:

Simplifying
1 + -3n = 0

Solving
1 + -3n = 0

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

Add '-1' to each side of the equation.
1 + -1 + -3n = 0 + -1

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

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

Divide each side by '-3'.
n = 0.3333333333

Simplifying
n = 0.3333333333
Solution
n = {-1.5, 0.3333333333}