Pre-Algebra Examples

$\frac{q^{2} - 11 q + 30}{4 q^{2} - 13 q^{3} - 35} \div \frac{q^{2} - 4 q - 12}{q^{2} - 3 q - 10}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{q^{2} - 11 q + 30}{4 q^{2} - 13 q^{3} - 35} \cdot \frac{q^{2} - 3 q - 10}{q^{2} - 4 q - 12}$

Step 2

Factor $q^{2} - 11 q + 30$ 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 $30$ and whose sum is $- 11$ .

$- 6, - 5$

Step 2.2

Write the factored form using these integers.

$\frac{(q - 6) (q - 5)}{4 q^{2} - 13 q^{3} - 35} \cdot \frac{q^{2} - 3 q - 10}{q^{2} - 4 q - 12}$

Step 3

Reorder terms.

$\frac{(q - 6) (q - 5)}{- 13 q^{3} + 4 q^{2} - 35} \cdot \frac{q^{2} - 3 q - 10}{q^{2} - 4 q - 12}$

Step 4

Factor $q^{2} - 3 q - 10$ using the AC method.

Tap for more steps...

Step 4.1

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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 4.2

Write the factored form using these integers.

$\frac{(q - 6) (q - 5)}{- 13 q^{3} + 4 q^{2} - 35} \cdot \frac{(q - 5) (q + 2)}{q^{2} - 4 q - 12}$

Step 5

Factor $q^{2} - 4 q - 12$ 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 $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Step 5.2

Write the factored form using these integers.

$\frac{(q - 6) (q - 5)}{- 13 q^{3} + 4 q^{2} - 35} \cdot \frac{(q - 5) (q + 2)}{(q - 6) (q + 2)}$

Step 6

Cancel the common factor of $q - 6$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

$\frac{(q - 6) (q - 5)}{- 13 q^{3} + 4 q^{2} - 35} \cdot \frac{(q - 5) (q + 2)}{(q - 6) (q + 2)}$

Step 6.2

Rewrite the expression.

$\frac{q - 5}{- 13 q^{3} + 4 q^{2} - 35} \cdot \frac{(q - 5) (q + 2)}{q + 2}$

Step 7

Multiply $\frac{q - 5}{- 13 q^{3} + 4 q^{2} - 35}$ by $\frac{(q - 5) (q + 2)}{q + 2}$ .

$\frac{(q - 5) ((q - 5) (q + 2))}{(- 13 q^{3} + 4 q^{2} - 35) (q + 2)}$

Step 8

Raise $q - 5$ to the power of $1$ .

$\frac{{(q - 5)}^{1} (q - 5) (q + 2)}{(- 13 q^{3} + 4 q^{2} - 35) (q + 2)}$

Step 9

Raise $q - 5$ to the power of $1$ .

$\frac{{(q - 5)}^{1} {(q - 5)}^{1} (q + 2)}{(- 13 q^{3} + 4 q^{2} - 35) (q + 2)}$

Step 10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(q - 5)}^{1 + 1} (q + 2)}{(- 13 q^{3} + 4 q^{2} - 35) (q + 2)}$

Step 11

Add $1$ and $1$ .

$\frac{{(q - 5)}^{2} (q + 2)}{(- 13 q^{3} + 4 q^{2} - 35) (q + 2)}$

Step 12

Cancel the common factor of $q + 2$ .

Tap for more steps...

Step 12.1

Cancel the common factor.

$\frac{{(q - 5)}^{2} (q + 2)}{(- 13 q^{3} + 4 q^{2} - 35) (q + 2)}$

Step 12.2

Rewrite the expression.

$\frac{{(q - 5)}^{2}}{- 13 q^{3} + 4 q^{2} - 35}$

Step 13

Factor $- 1$ out of $- 13 q^{3}$ .

$\frac{{(q - 5)}^{2}}{- (13 q^{3}) + 4 q^{2} - 35}$

Step 14

Factor $- 1$ out of $4 q^{2}$ .

$\frac{{(q - 5)}^{2}}{- (13 q^{3}) - (- 4 q^{2}) - 35}$

Step 15

Factor $- 1$ out of $- (13 q^{3}) - (- 4 q^{2})$ .

$\frac{{(q - 5)}^{2}}{- (13 q^{3} - 4 q^{2}) - 35}$

Step 16

Rewrite $- 35$ as $- 1 (35)$ .

$\frac{{(q - 5)}^{2}}{- (13 q^{3} - 4 q^{2}) - 1 (35)}$

Step 17

Factor $- 1$ out of $- (13 q^{3} - 4 q^{2}) - 1 (35)$ .

$\frac{{(q - 5)}^{2}}{- (13 q^{3} - 4 q^{2} + 35)}$

Step 18

Rewrite $- (13 q^{3} - 4 q^{2} + 35)$ as $- 1 (13 q^{3} - 4 q^{2} + 35)$ .

$\frac{{(q - 5)}^{2}}{- 1 (13 q^{3} - 4 q^{2} + 35)}$

Step 19

Move the negative in front of the fraction.

$- \frac{{(q - 5)}^{2}}{13 q^{3} - 4 q^{2} + 35}$

Step 20

Rewrite ${(q - 5)}^{2}$ as $(q - 5) (q - 5)$ .

$- \frac{(q - 5) (q - 5)}{13 q^{3} - 4 q^{2} + 35}$

Step 21

Expand $(q - 5) (q - 5)$ using the FOIL Method.

Tap for more steps...

Step 21.1

Apply the distributive property.

$- \frac{q (q - 5) - 5 (q - 5)}{13 q^{3} - 4 q^{2} + 35}$

Step 21.2

Apply the distributive property.

$- \frac{q \cdot q + q \cdot - 5 - 5 (q - 5)}{13 q^{3} - 4 q^{2} + 35}$

Step 21.3

Apply the distributive property.

$- \frac{q \cdot q + q \cdot - 5 - 5 q - 5 \cdot - 5}{13 q^{3} - 4 q^{2} + 35}$

Step 22

Simplify and combine like terms.

Tap for more steps...

Step 22.1

Simplify each term.

Tap for more steps...

Step 22.1.1

Multiply $q$ by $q$ .

$- \frac{q^{2} + q \cdot - 5 - 5 q - 5 \cdot - 5}{13 q^{3} - 4 q^{2} + 35}$

Step 22.1.2

Move $- 5$ to the left of $q$ .

$- \frac{q^{2} - 5 \cdot q - 5 q - 5 \cdot - 5}{13 q^{3} - 4 q^{2} + 35}$

Step 22.1.3

Multiply $- 5$ by $- 5$ .

$- \frac{q^{2} - 5 q - 5 q + 25}{13 q^{3} - 4 q^{2} + 35}$

Step 22.2

Subtract $5 q$ from $- 5 q$ .

$- \frac{q^{2} - 10 q + 25}{13 q^{3} - 4 q^{2} + 35}$

Step 23

Split the fraction $\frac{q^{2} - 10 q + 25}{13 q^{3} - 4 q^{2} + 35}$ into two fractions.

$- (\frac{q^{2} - 10 q}{13 q^{3} - 4 q^{2} + 35} + \frac{25}{13 q^{3} - 4 q^{2} + 35})$

Step 24

Split the fraction $\frac{q^{2} - 10 q}{13 q^{3} - 4 q^{2} + 35}$ into two fractions.

$- (\frac{q^{2}}{13 q^{3} - 4 q^{2} + 35} + \frac{- 10 q}{13 q^{3} - 4 q^{2} + 35} + \frac{25}{13 q^{3} - 4 q^{2} + 35})$

Step 25

Move the negative in front of the fraction.

$- (\frac{q^{2}}{13 q^{3} - 4 q^{2} + 35} - \frac{10 q}{13 q^{3} - 4 q^{2} + 35} + \frac{25}{13 q^{3} - 4 q^{2} + 35})$