Find Where Undefined/Discontinuous (y^2+6y+9)/(y^2-y-12)

$\frac{y^{2} + 6 y + 9}{y^{2} - y - 12}$

Step 1

Set the denominator in $\frac{y^{2} + 6 y + 9}{y^{2} - y - 12}$ equal to $0$ to find where the expression is undefined.

$y^{2} - y - 12 = 0$

Step 2

Solve for $y$ .

Factor $y^{2} - y - 12$ 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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Write the factored form using these integers.

$(y - 4) (y + 3) = 0$

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

$y - 4 = 0$

$y + 3 = 0$

Set $y - 4$ equal to $0$ and solve for $y$ .

Set $y - 4$ equal to $0$ .

$y - 4 = 0$

Add $4$ to both sides of the equation.

$y = 4$

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

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

$y + 3 = 0$

Subtract $3$ from both sides of the equation.

$y = - 3$

The final solution is all the values that make $(y - 4) (y + 3) = 0$ true.

$y = 4, - 3$

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 $0$ .

$y = - 3, y = 4$

Step 4

Find Where Undefined/Discontinuous (y^2+6y+9)/(y^2-y-12)

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