Find Where Undefined/Discontinuous (z^2+3z-54)/(z^2-9z+18)

$\frac{z^{2} + 3 z - 54}{z^{2} - 9 z + 18}$

Step 1

Set the denominator in $\frac{z^{2} + 3 z - 54}{z^{2} - 9 z + 18}$ equal to $0$ to find where the expression is undefined.

$z^{2} - 9 z + 18 = 0$

Step 2

Solve for $z$ .

Factor $z^{2} - 9 z + 18$ 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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Write the factored form using these integers.

$(z - 6) (z - 3) = 0$

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

$z - 6 = 0$

$z - 3 = 0$

Set $z - 6$ equal to $0$ and solve for $z$ .

Set $z - 6$ equal to $0$ .

$z - 6 = 0$

Add $6$ to both sides of the equation.

$z = 6$

Set $z - 3$ equal to $0$ and solve for $z$ .

Set $z - 3$ equal to $0$ .

$z - 3 = 0$

Add $3$ to both sides of the equation.

$z = 3$

The final solution is all the values that make $(z - 6) (z - 3) = 0$ true.

$z = 6, 3$

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 $0$ .

$z = 3, z = 6$

Step 4

Find Where Undefined/Discontinuous (z^2+3z-54)/(z^2-9z+18)

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