Find the Excluded Values (-3n+12)/(84n^2+76n+16)

$\frac{- 3 n + 12}{84 n^{2} + 76 n + 16}$

Set the denominator in $\frac{- 3 n + 12}{84 n^{2} + 76 n + 16}$ equal to $0$ to find where the expression is undefined.

$84 n^{2} + 76 n + 16 = 0$

Solve for $n$ .

Factor the left side of the equation.

Factor $4$ out of $84 n^{2} + 76 n + 16$ .

Factor $4$ out of $84 n^{2}$ .

$4 (21 n^{2}) + 76 n + 16 = 0$

Factor $4$ out of $76 n$ .

$4 (21 n^{2}) + 4 (19 n) + 16 = 0$

Factor $4$ out of $16$ .

$4 (21 n^{2}) + 4 (19 n) + 4 (4) = 0$

Factor $4$ out of $4 (21 n^{2}) + 4 (19 n)$ .

$4 (21 n^{2} + 19 n) + 4 (4) = 0$

Factor $4$ out of $4 (21 n^{2} + 19 n) + 4 (4)$ .

$4 (21 n^{2} + 19 n + 4) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 21 \cdot 4 = 84$ and whose sum is $b = 19$ .

Factor $19$ out of $19 n$ .

$4 (21 n^{2} + 19 (n) + 4) = 0$

Rewrite $19$ as $7$ plus $12$

$4 (21 n^{2} + (7 + 12) n + 4) = 0$

Apply the distributive property.

$4 (21 n^{2} + 7 n + 12 n + 4) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 ((21 n^{2} + 7 n) + 12 n + 4) = 0$

Factor out the greatest common factor (GCF) from each group.

$4 (7 n (3 n + 1) + 4 (3 n + 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 n + 1$ .

$4 ((3 n + 1) (7 n + 4)) = 0$

Remove unnecessary parentheses.

$4 (3 n + 1) (7 n + 4) = 0$

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

$3 n + 1 = 0$

$7 n + 4 = 0$

Set $3 n + 1$ equal to $0$ and solve for $n$ .

Set $3 n + 1$ equal to $0$ .

$3 n + 1 = 0$

Solve $3 n + 1 = 0$ for $n$ .

Subtract $1$ from both sides of the equation.

$3 n = - 1$

Divide each term in $3 n = - 1$ by $3$ and simplify.

Divide each term in $3 n = - 1$ by $3$ .

$\frac{3 n}{3} = \frac{- 1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 n}{3} = \frac{- 1}{3}$

Divide $n$ by $1$ .

$n = \frac{- 1}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$n = - \frac{1}{3}$

Set $7 n + 4$ equal to $0$ and solve for $n$ .

Set $7 n + 4$ equal to $0$ .

$7 n + 4 = 0$

Solve $7 n + 4 = 0$ for $n$ .

Subtract $4$ from both sides of the equation.

$7 n = - 4$

Divide each term in $7 n = - 4$ by $7$ and simplify.

Divide each term in $7 n = - 4$ by $7$ .

$\frac{7 n}{7} = \frac{- 4}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 n}{7} = \frac{- 4}{7}$

Divide $n$ by $1$ .

$n = \frac{- 4}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$n = - \frac{4}{7}$

The final solution is all the values that make $4 (3 n + 1) (7 n + 4) = 0$ true.

$n = - \frac{1}{3}, - \frac{4}{7}$

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 $0$ .

$n = - \frac{4}{7}, n = - \frac{1}{3}$

Find the Excluded Values (-3n+12)/(84n^2+76n+16)

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