Find Where Undefined/Discontinuous ((z^2-36)/(20z^2-25z))÷((4z^2-4z-120)/(5z^3))

$\frac{z^{2} - 36}{20 z^{2} - 25 z} \div \frac{4 z^{2} - 4 z - 120}{5 z^{3}}$

Step 1

Set the denominator in $\frac{z^{2} - 36}{20 z^{2} - 25 z}$ equal to $0$ to find where the expression is undefined.

$20 z^{2} - 25 z = 0$

Step 2

Solve for $z$ .

Factor $5 z$ out of $20 z^{2} - 25 z$ .

Factor $5 z$ out of $20 z^{2}$ .

$5 z (4 z) - 25 z = 0$

Factor $5 z$ out of $- 25 z$ .

$5 z (4 z) + 5 z (- 5) = 0$

Factor $5 z$ out of $5 z (4 z) + 5 z (- 5)$ .

$5 z (4 z - 5) = 0$

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

$z = 0$

$4 z - 5 = 0$

Set $z$ equal to $0$ .

$z = 0$

Set $4 z - 5$ equal to $0$ and solve for $z$ .

Set $4 z - 5$ equal to $0$ .

$4 z - 5 = 0$

Solve $4 z - 5 = 0$ for $z$ .

Add $5$ to both sides of the equation.

$4 z = 5$

Divide each term in $4 z = 5$ by $4$ and simplify.

Divide each term in $4 z = 5$ by $4$ .

$\frac{4 z}{4} = \frac{5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 z}{4} = \frac{5}{4}$

Divide $z$ by $1$ .

$z = \frac{5}{4}$

The final solution is all the values that make $5 z (4 z - 5) = 0$ true.

$z = 0, \frac{5}{4}$

Step 3

Set the denominator in $\frac{4 z^{2} - 4 z - 120}{5 z^{3}}$ equal to $0$ to find where the expression is undefined.

$5 z^{3} = 0$

Step 4

Solve for $z$ .

Divide each term in $5 z^{3} = 0$ by $5$ and simplify.

Divide each term in $5 z^{3} = 0$ by $5$ .

$\frac{5 z^{3}}{5} = \frac{0}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 z^{3}}{5} = \frac{0}{5}$

Divide $z^{3}$ by $1$ .

$z^{3} = \frac{0}{5}$

Simplify the right side.

Divide $0$ by $5$ .

$z^{3} = 0$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$z = \sqrt[3]{0}$

Simplify $\sqrt[3]{0}$ .

Rewrite $0$ as $0^{3}$ .

$z = \sqrt[3]{0^{3}}$

Pull terms out from under the radical, assuming real numbers.

$z = 0$

Step 5

Set the denominator in $\frac{\frac{z^{2} - 36}{20 z^{2} - 25 z}}{\frac{4 z^{2} - 4 z - 120}{5 z^{3}}}$ equal to $0$ to find where the expression is undefined.

$\frac{4 z^{2} - 4 z - 120}{5 z^{3}} = 0$

Step 6

Solve for $z$ .

Set the numerator equal to zero.

$4 z^{2} - 4 z - 120 = 0$

Solve the equation for $z$ .

Factor the left side of the equation.

Factor $4$ out of $4 z^{2} - 4 z - 120$ .

Factor $4$ out of $4 z^{2}$ .

$4 (z^{2}) - 4 z - 120 = 0$

Factor $4$ out of $- 4 z$ .

$4 (z^{2}) + 4 (- z) - 120 = 0$

Factor $4$ out of $- 120$ .

$4 (z^{2}) + 4 (- z) + 4 (- 30) = 0$

Factor $4$ out of $4 (z^{2}) + 4 (- z)$ .

$4 (z^{2} - z) + 4 (- 30) = 0$

Factor $4$ out of $4 (z^{2} - z) + 4 (- 30)$ .

$4 (z^{2} - z - 30) = 0$

Factor.

Factor $z^{2} - z - 30$ 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 $- 30$ and whose sum is $- 1$ .

$- 6, 5$

Write the factored form using these integers.

$4 ((z - 6) (z + 5)) = 0$

Remove unnecessary parentheses.

$4 (z - 6) (z + 5) = 0$

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

$z - 6 = 0$

$z + 5 = 0$

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 $z + 5$ equal to $0$ and solve for $z$ .

Set $z + 5$ equal to $0$ .

$z + 5 = 0$

Subtract $5$ from both sides of the equation.

$z = - 5$

The final solution is all the values that make $4 (z - 6) (z + 5) = 0$ true.

$z = 6, - 5$

Step 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 $0$ .

$z = - 5, z = 0, z = \frac{5}{4}, z = 6$

Step 8

Find Where Undefined/Discontinuous ((z^2-36)/(20z^2-25z))÷((4z^2-4z-120)/(5z^3))

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