(n-3)(n-4)=42

Solution for (n-3)(n-4)=42 equation:

Simplifying
(n + -3)(n + -4) = 42

Reorder the terms:
(-3 + n)(n + -4) = 42

Reorder the terms:
(-3 + n)(-4 + n) = 42

Multiply (-3 + n) * (-4 + n)
(-3(-4 + n) + n(-4 + n)) = 42
((-4 * -3 + n * -3) + n(-4 + n)) = 42
((12 + -3n) + n(-4 + n)) = 42
(12 + -3n + (-4 * n + n * n)) = 42
(12 + -3n + (-4n + n2)) = 42

Combine like terms: -3n + -4n = -7n
(12 + -7n + n2) = 42

Solving
12 + -7n + n2 = 42

Solving for variable 'n'.

Reorder the terms:
12 + -42 + -7n + n2 = 42 + -42

Combine like terms: 12 + -42 = -30
-30 + -7n + n2 = 42 + -42

Combine like terms: 42 + -42 = 0
-30 + -7n + n2 = 0

Factor a trinomial.
(-3 + -1n)(10 + -1n) = 0

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

Simplifying
-3 + -1n = 0

Solving
-3 + -1n = 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 + -1n = 0 + 3

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

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

Divide each side by '-1'.
n = -3

Simplifying
n = -3
Subproblem 2
Set the factor '(10 + -1n)' equal to zero and attempt to solve:

Simplifying
10 + -1n = 0

Solving
10 + -1n = 0

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

Add '-10' to each side of the equation.
10 + -10 + -1n = 0 + -10

Combine like terms: 10 + -10 = 0
0 + -1n = 0 + -10
-1n = 0 + -10

Combine like terms: 0 + -10 = -10
-1n = -10

Divide each side by '-1'.
n = 10

Simplifying
n = 10
Solution
n = {-3, 10}