Divide ((z^2+11z+18)/(z^2+14z+45))÷((z^2+2z)/(z^2+10z+25))

$\frac{z^{2} + 11 z + 18}{z^{2} + 14 z + 45} \div \frac{z^{2} + 2 z}{z^{2} + 10 z + 25}$

To divide by a fraction, multiply by its reciprocal.

$\frac{z^{2} + 11 z + 18}{z^{2} + 14 z + 45} \cdot \frac{z^{2} + 10 z + 25}{z^{2} + 2 z}$

Factor $z^{2} + 11 z + 18$ using the AC method.

Tap for more steps…

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 $11$ .

$2, 9$

Write the factored form using these integers.

$\frac{(z + 2) (z + 9)}{z^{2} + 14 z + 45} \cdot \frac{z^{2} + 10 z + 25}{z^{2} + 2 z}$

Factor $z^{2} + 14 z + 45$ using the AC method.

Tap for more steps…

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 $45$ and whose sum is $14$ .

$5, 9$

Write the factored form using these integers.

$\frac{(z + 2) (z + 9)}{(z + 5) (z + 9)} \cdot \frac{z^{2} + 10 z + 25}{z^{2} + 2 z}$

Cancel the common factor of $z + 9$ .

Tap for more steps…

Cancel the common factor.

$\frac{(z + 2) (z + 9)}{(z + 5) (z + 9)} \cdot \frac{z^{2} + 10 z + 25}{z^{2} + 2 z}$

Rewrite the expression.

$\frac{z + 2}{z + 5} \cdot \frac{z^{2} + 10 z + 25}{z^{2} + 2 z}$

Factor using the perfect square rule.

Tap for more steps…

Rewrite $25$ as $5^{2}$ .

$\frac{z + 2}{z + 5} \cdot \frac{z^{2} + 10 z + 5^{2}}{z^{2} + 2 z}$

Check the middle term by multiplying $2 a b$ and compare this result with the middle term in the original expression.

$2 a b = 2 \cdot z \cdot 5$

Simplify.

$2 a b = 10 z$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = z$ and $b = 5$ .

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{z^{2} + 2 z}$

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

Tap for more steps…

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

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{z \cdot z + 2 z}$

Factor $z$ out of $2 z$ .

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{z \cdot z + z \cdot 2}$

Factor $z$ out of $z \cdot z + z \cdot 2$ .

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{z (z + 2)}$

Cancel the common factor of $z + 2$ .

Tap for more steps…

Factor $z + 2$ out of $z (z + 2)$ .

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{(z + 2) z}$

Cancel the common factor.

$\frac{z + 2}{z + 5} \cdot \frac{{(z + 5)}^{2}}{(z + 2) z}$

Rewrite the expression.

$\frac{1}{z + 5} \cdot \frac{{(z + 5)}^{2}}{z}$

Cancel the common factor of $z + 5$ .

Tap for more steps…

Factor $z + 5$ out of ${(z + 5)}^{2}$ .

$\frac{1}{z + 5} \cdot \frac{(z + 5) (z + 5)}{z}$

Cancel the common factor.

$\frac{1}{z + 5} \cdot \frac{(z + 5) (z + 5)}{z}$

Rewrite the expression.

$\frac{z + 5}{z}$

Split the fraction $\frac{z + 5}{z}$ into two fractions.

$\frac{z}{z} + \frac{5}{z}$

Cancel the common factor of $z$ .

Tap for more steps…

Cancel the common factor.

$\frac{z}{z} + \frac{5}{z}$

Divide $1$ by $1$ .

$1 + \frac{5}{z}$

Divide ((z^2+11z+18)/(z^2+14z+45))÷((z^2+2z)/(z^2+10z+25))

Download our
App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our
App from the store