Basic Math Examples

$\frac{z^{2} + 12 z + 35}{z^{2} + 15 z + 56} \div \frac{z^{2} + 5 z}{z^{2} + 3 z - 40}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{z^{2} + 12 z + 35}{z^{2} + 15 z + 56} \cdot \frac{z^{2} + 3 z - 40}{z^{2} + 5 z}$

Step 2

Factor $z^{2} + 12 z + 35$ using the AC method.

Tap for more steps...

Step 2.1

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 $35$ and whose sum is $12$ .

$5, 7$

Step 2.2

Write the factored form using these integers.

$\frac{(z + 5) (z + 7)}{z^{2} + 15 z + 56} \cdot \frac{z^{2} + 3 z - 40}{z^{2} + 5 z}$

Step 3

Factor $z^{2} + 15 z + 56$ using the AC method.

Tap for more steps...

Step 3.1

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 $56$ and whose sum is $15$ .

$7, 8$

Step 3.2

Write the factored form using these integers.

$\frac{(z + 5) (z + 7)}{(z + 7) (z + 8)} \cdot \frac{z^{2} + 3 z - 40}{z^{2} + 5 z}$

Step 4

Cancel the common factor of $z + 7$ .

Tap for more steps...

Step 4.1

Cancel the common factor.

$\frac{(z + 5) (z + 7)}{(z + 7) (z + 8)} \cdot \frac{z^{2} + 3 z - 40}{z^{2} + 5 z}$

Step 4.2

Rewrite the expression.

$\frac{z + 5}{z + 8} \cdot \frac{z^{2} + 3 z - 40}{z^{2} + 5 z}$

Step 5

Factor $z^{2} + 3 z - 40$ using the AC method.

Tap for more steps...

Step 5.1

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 $- 40$ and whose sum is $3$ .

$- 5, 8$

Step 5.2

Write the factored form using these integers.

$\frac{z + 5}{z + 8} \cdot \frac{(z - 5) (z + 8)}{z^{2} + 5 z}$

Step 6

Simplify terms.

Tap for more steps...

Step 6.1

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

Tap for more steps...

Step 6.1.1

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

$\frac{z + 5}{z + 8} \cdot \frac{(z - 5) (z + 8)}{z \cdot z + 5 z}$

Step 6.1.2

Factor $z$ out of $5 z$ .

$\frac{z + 5}{z + 8} \cdot \frac{(z - 5) (z + 8)}{z \cdot z + z \cdot 5}$

Step 6.1.3

Factor $z$ out of $z \cdot z + z \cdot 5$ .

$\frac{z + 5}{z + 8} \cdot \frac{(z - 5) (z + 8)}{z (z + 5)}$

Step 6.2

Cancel the common factor of $z + 5$ .

Tap for more steps...

Step 6.2.1

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

$\frac{z + 5}{z + 8} \cdot \frac{(z - 5) (z + 8)}{(z + 5) z}$

Step 6.2.2

Cancel the common factor.

$\frac{z + 5}{z + 8} \cdot \frac{(z - 5) (z + 8)}{(z + 5) z}$

Step 6.2.3

Rewrite the expression.

$\frac{1}{z + 8} \cdot \frac{(z - 5) (z + 8)}{z}$

Step 6.3

Cancel the common factor of $z + 8$ .

Tap for more steps...

Step 6.3.1

Factor $z + 8$ out of $(z - 5) (z + 8)$ .

$\frac{1}{z + 8} \cdot \frac{(z + 8) (z - 5)}{z}$

Step 6.3.2

Cancel the common factor.

$\frac{1}{z + 8} \cdot \frac{(z + 8) (z - 5)}{z}$

Step 6.3.3

Rewrite the expression.

$\frac{z - 5}{z}$