Find Where Undefined/Discontinuous (z^2+8z+16)/(z^2-5z-36)

$\frac{z^{2} + 8 z + 16}{z^{2} - 5 z - 36}$

Step 1

Set the denominator in $\frac{z^{2} + 8 z + 16}{z^{2} - 5 z - 36}$ equal to $0$ to find where the expression is undefined.

$z^{2} - 5 z - 36 = 0$

Step 2

Solve for $z$ .

Factor $z^{2} - 5 z - 36$ 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 $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Write the factored form using these integers.

$(z - 9) (z + 4) = 0$

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

$z - 9 = 0$

$z + 4 = 0$

Set $z - 9$ equal to $0$ and solve for $z$ .

Set $z - 9$ equal to $0$ .

$z - 9 = 0$

Add $9$ to both sides of the equation.

$z = 9$

Set $z + 4$ equal to $0$ and solve for $z$ .

Set $z + 4$ equal to $0$ .

$z + 4 = 0$

Subtract $4$ from both sides of the equation.

$z = - 4$

The final solution is all the values that make $(z - 9) (z + 4) = 0$ true.

$z = 9, - 4$

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 $0$ .

$z = - 4, z = 9$

Step 4

Find Where Undefined/Discontinuous (z^2+8z+16)/(z^2-5z-36)

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