(n-5)(n+2)=78

Solution for (n-5)(n+2)=78 equation:

Simplifying
(n + -5)(n + 2) = 78

Reorder the terms:
(-5 + n)(n + 2) = 78

Reorder the terms:
(-5 + n)(2 + n) = 78

Multiply (-5 + n) * (2 + n)
(-5(2 + n) + n(2 + n)) = 78
((2 * -5 + n * -5) + n(2 + n)) = 78
((-10 + -5n) + n(2 + n)) = 78
(-10 + -5n + (2 * n + n * n)) = 78
(-10 + -5n + (2n + n2)) = 78

Combine like terms: -5n + 2n = -3n
(-10 + -3n + n2) = 78

Solving
-10 + -3n + n2 = 78

Solving for variable 'n'.

Reorder the terms:
-10 + -78 + -3n + n2 = 78 + -78

Combine like terms: -10 + -78 = -88
-88 + -3n + n2 = 78 + -78

Combine like terms: 78 + -78 = 0
-88 + -3n + n2 = 0

Factor a trinomial.
(-8 + -1n)(11 + -1n) = 0

Subproblem 1
Set the factor '(-8 + -1n)' equal to zero and attempt to solve:

Simplifying
-8 + -1n = 0

Solving
-8 + -1n = 0

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

Add '8' to each side of the equation.
-8 + 8 + -1n = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -1n = 0 + 8
-1n = 0 + 8

Combine like terms: 0 + 8 = 8
-1n = 8

Divide each side by '-1'.
n = -8

Simplifying
n = -8
Subproblem 2
Set the factor '(11 + -1n)' equal to zero and attempt to solve:

Simplifying
11 + -1n = 0

Solving
11 + -1n = 0

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

Add '-11' to each side of the equation.
11 + -11 + -1n = 0 + -11

Combine like terms: 11 + -11 = 0
0 + -1n = 0 + -11
-1n = 0 + -11

Combine like terms: 0 + -11 = -11
-1n = -11

Divide each side by '-1'.
n = 11

Simplifying
n = 11
Solution
n = {-8, 11}