Basic Math Examples

$\frac{4 y^{2} + 8 y - 252}{y^{2} - 9 y + 14} \div \frac{5 y^{2} + 48 y + 27}{15 y^{2} - 21 y - 18}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 y^{2} + 8 y - 252}{y^{2} - 9 y + 14} \cdot \frac{15 y^{2} - 21 y - 18}{5 y^{2} + 48 y + 27}$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Factor $4$ out of $4 y^{2} + 8 y - 252$ .

Tap for more steps...

Step 2.1.1

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

$\frac{4 (y^{2}) + 8 y - 252}{y^{2} - 9 y + 14} \cdot \frac{15 y^{2} - 21 y - 18}{5 y^{2} + 48 y + 27}$

Step 2.1.2

Factor $4$ out of $8 y$ .

$\frac{4 (y^{2}) + 4 (2 y) - 252}{y^{2} - 9 y + 14} \cdot \frac{15 y^{2} - 21 y - 18}{5 y^{2} + 48 y + 27}$

Step 2.1.3

Factor $4$ out of $- 252$ .

$\frac{4 y^{2} + 4 (2 y) + 4 \cdot - 63}{y^{2} - 9 y + 14} \cdot \frac{15 y^{2} - 21 y - 18}{5 y^{2} + 48 y + 27}$

Step 2.1.4

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

$\frac{4 (y^{2} + 2 y) + 4 \cdot - 63}{y^{2} - 9 y + 14} \cdot \frac{15 y^{2} - 21 y - 18}{5 y^{2} + 48 y + 27}$

Step 2.1.5

Factor $4$ out of $4 (y^{2} + 2 y) + 4 \cdot - 63$ .

$\frac{4 (y^{2} + 2 y - 63)}{y^{2} - 9 y + 14} \cdot \frac{15 y^{2} - 21 y - 18}{5 y^{2} + 48 y + 27}$

Step 2.2

Factor $y^{2} + 2 y - 63$ using the AC method.

Tap for more steps...

Step 2.2.1

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 $- 63$ and whose sum is $2$ .

$- 7, 9$

Step 2.2.2

Write the factored form using these integers.

$\frac{4 ((y - 7) (y + 9))}{y^{2} - 9 y + 14} \cdot \frac{15 y^{2} - 21 y - 18}{5 y^{2} + 48 y + 27}$

$\frac{4 (y - 7) (y + 9)}{y^{2} - 9 y + 14} \cdot \frac{15 y^{2} - 21 y - 18}{5 y^{2} + 48 y + 27}$

Step 3

Factor $y^{2} - 9 y + 14$ 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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Step 3.2

Write the factored form using these integers.

$\frac{4 (y - 7) (y + 9)}{(y - 7) (y - 2)} \cdot \frac{15 y^{2} - 21 y - 18}{5 y^{2} + 48 y + 27}$

Step 4

Cancel the common factor of $y - 7$ .

Tap for more steps...

Step 4.1

Cancel the common factor.

$\frac{4 (y - 7) (y + 9)}{(y - 7) (y - 2)} \cdot \frac{15 y^{2} - 21 y - 18}{5 y^{2} + 48 y + 27}$

Step 4.2

Rewrite the expression.

$\frac{4 (y + 9)}{y - 2} \cdot \frac{15 y^{2} - 21 y - 18}{5 y^{2} + 48 y + 27}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Factor $3$ out of $15 y^{2} - 21 y - 18$ .

Tap for more steps...

Step 5.1.1

Factor $3$ out of $15 y^{2}$ .

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y^{2}) - 21 y - 18}{5 y^{2} + 48 y + 27}$

Step 5.1.2

Factor $3$ out of $- 21 y$ .

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y^{2}) + 3 (- 7 y) - 18}{5 y^{2} + 48 y + 27}$

Step 5.1.3

Factor $3$ out of $- 18$ .

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y^{2}) + 3 (- 7 y) + 3 (- 6)}{5 y^{2} + 48 y + 27}$

Step 5.1.4

Factor $3$ out of $3 (5 y^{2}) + 3 (- 7 y)$ .

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y^{2} - 7 y) + 3 (- 6)}{5 y^{2} + 48 y + 27}$

Step 5.1.5

Factor $3$ out of $3 (5 y^{2} - 7 y) + 3 (- 6)$ .

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y^{2} - 7 y - 6)}{5 y^{2} + 48 y + 27}$

Step 5.2

Factor by grouping.

Tap for more steps...

Step 5.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 6 = - 30$ and whose sum is $b = - 7$ .

Tap for more steps...

Step 5.2.1.1

Factor $- 7$ out of $- 7 y$ .

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y^{2} - 7 (y) - 6)}{5 y^{2} + 48 y + 27}$

Step 5.2.1.2

Rewrite $- 7$ as $3$ plus $- 10$

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y^{2} + (3 - 10) y - 6)}{5 y^{2} + 48 y + 27}$

Step 5.2.1.3

Apply the distributive property.

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y^{2} + 3 y - 10 y - 6)}{5 y^{2} + 48 y + 27}$

Step 5.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 5.2.2.1

Group the first two terms and the last two terms.

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 ((5 y^{2} + 3 y) - 10 y - 6)}{5 y^{2} + 48 y + 27}$

Step 5.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (y (5 y + 3) - 2 (5 y + 3))}{5 y^{2} + 48 y + 27}$

Step 5.2.3

Factor the polynomial by factoring out the greatest common factor, $5 y + 3$ .

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 ((5 y + 3) (y - 2))}{5 y^{2} + 48 y + 27}$

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y + 3) (y - 2)}{5 y^{2} + 48 y + 27}$

Step 6

Factor by grouping.

Tap for more steps...

Step 6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 27 = 135$ and whose sum is $b = 48$ .

Tap for more steps...

Step 6.1.1

Factor $48$ out of $48 y$ .

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y + 3) (y - 2)}{5 y^{2} + 48 (y) + 27}$

Step 6.1.2

Rewrite $48$ as $3$ plus $45$

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y + 3) (y - 2)}{5 y^{2} + (3 + 45) y + 27}$

Step 6.1.3

Apply the distributive property.

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y + 3) (y - 2)}{5 y^{2} + 3 y + 45 y + 27}$

Step 6.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 6.2.1

Group the first two terms and the last two terms.

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y + 3) (y - 2)}{(5 y^{2} + 3 y) + 45 y + 27}$

Step 6.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y + 3) (y - 2)}{y (5 y + 3) + 9 (5 y + 3)}$

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $5 y + 3$ .

$\frac{4 (y + 9)}{y - 2} \cdot \frac{3 (5 y + 3) (y - 2)}{(5 y + 3) (y + 9)}$

Step 7

Cancel the common factor of $y + 9$ .

Tap for more steps...

Step 7.1

Factor $y + 9$ out of $4 (y + 9)$ .

$\frac{(y + 9) \cdot 4}{y - 2} \cdot \frac{3 (5 y + 3) (y - 2)}{(5 y + 3) (y + 9)}$

Step 7.2

Factor $y + 9$ out of $(5 y + 3) (y + 9)$ .

$\frac{(y + 9) \cdot 4}{y - 2} \cdot \frac{3 (5 y + 3) (y - 2)}{(y + 9) (5 y + 3)}$

Step 7.3

Cancel the common factor.

$\frac{(y + 9) \cdot 4}{y - 2} \cdot \frac{3 (5 y + 3) (y - 2)}{(y + 9) (5 y + 3)}$

Step 7.4

Rewrite the expression.

$\frac{4}{y - 2} \cdot \frac{3 (5 y + 3) (y - 2)}{5 y + 3}$

Step 8

Cancel the common factor of $y - 2$ .

Tap for more steps...

Step 8.1

Factor $y - 2$ out of $3 (5 y + 3) (y - 2)$ .

$\frac{4}{y - 2} \cdot \frac{(y - 2) (3 (5 y + 3))}{5 y + 3}$

Step 8.2

Cancel the common factor.

$\frac{4}{y - 2} \cdot \frac{(y - 2) (3 (5 y + 3))}{5 y + 3}$

Step 8.3

Rewrite the expression.

$4 \frac{3 (5 y + 3)}{5 y + 3}$

Step 9

Combine $4$ and $\frac{3 (5 y + 3)}{5 y + 3}$ .

$\frac{4 (3 (5 y + 3))}{5 y + 3}$

Step 10

Multiply $3$ by $4$ .

$\frac{12 (5 y + 3)}{5 y + 3}$

Step 11

Cancel the common factor of $5 y + 3$ .

Tap for more steps...

Step 11.1

Cancel the common factor.

$\frac{12 (5 y + 3)}{5 y + 3}$

Step 11.2

Divide $12$ by $1$ .

$12$