Find Where Undefined/Discontinuous (n^2+6n+9)/(3n^2+15n+18)

$\frac{n^{2} + 6 n + 9}{3 n^{2} + 15 n + 18}$

Step 1

Set the denominator in $\frac{n^{2} + 6 n + 9}{3 n^{2} + 15 n + 18}$ equal to $0$ to find where the expression is undefined.

$3 n^{2} + 15 n + 18 = 0$

Step 2

Solve for $n$ .

Factor the left side of the equation.

Factor $3$ out of $3 n^{2} + 15 n + 18$ .

Factor $3$ out of $3 n^{2}$ .

$3 (n^{2}) + 15 n + 18 = 0$

Factor $3$ out of $15 n$ .

$3 (n^{2}) + 3 (5 n) + 18 = 0$

Factor $3$ out of $18$ .

$3 n^{2} + 3 (5 n) + 3 \cdot 6 = 0$

Factor $3$ out of $3 n^{2} + 3 (5 n)$ .

$3 (n^{2} + 5 n) + 3 \cdot 6 = 0$

Factor $3$ out of $3 (n^{2} + 5 n) + 3 \cdot 6$ .

$3 (n^{2} + 5 n + 6) = 0$

Factor.

Factor $n^{2} + 5 n + 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $5$ .

$2, 3$

Write the factored form using these integers.

$3 ((n + 2) (n + 3)) = 0$

Remove unnecessary parentheses.

$3 (n + 2) (n + 3) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$n + 2 = 0$

$n + 3 = 0$

Set $n + 2$ equal to $0$ and solve for $n$ .

Set $n + 2$ equal to $0$ .

$n + 2 = 0$

Subtract $2$ from both sides of the equation.

$n = - 2$

Set $n + 3$ equal to $0$ and solve for $n$ .

Set $n + 3$ equal to $0$ .

$n + 3 = 0$

Subtract $3$ from both sides of the equation.

$n = - 3$

The final solution is all the values that make $3 (n + 2) (n + 3) = 0$ true.

$n = - 2, - 3$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$n = - 3, n = - 2$

Step 4

Find Where Undefined/Discontinuous (n^2+6n+9)/(3n^2+15n+18)

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