Find Where Undefined/Discontinuous (187+6z-z^2)/(z^2-21z+68)

$\frac{187 + 6 z - z^{2}}{z^{2} - 21 z + 68}$

Step 1

Set the denominator in $\frac{187 + 6 z - z^{2}}{z^{2} - 21 z + 68}$ equal to $0$ to find where the expression is undefined.

$z^{2} - 21 z + 68 = 0$

Step 2

Solve for $z$ .

Factor $z^{2} - 21 z + 68$ 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 $68$ and whose sum is $- 21$ .

$- 17, - 4$

Write the factored form using these integers.

$(z - 17) (z - 4) = 0$

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

$z - 17 = 0$

$z - 4 = 0$

Set $z - 17$ equal to $0$ and solve for $z$ .

Set $z - 17$ equal to $0$ .

$z - 17 = 0$

Add $17$ to both sides of the equation.

$z = 17$

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

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

$z - 4 = 0$

Add $4$ to both sides of the equation.

$z = 4$

The final solution is all the values that make $(z - 17) (z - 4) = 0$ true.

$z = 17, 4$

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 $0$ .

$z = 4, z = 17$

Step 4

Find Where Undefined/Discontinuous (187+6z-z^2)/(z^2-21z+68)

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