Basic Math Examples

$\frac{4 - 9 y^{2}}{3 y^{2} + 4 y - 4} \div \frac{3 y^{2} + 5 y + 3}{y^{2} + y - 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.2

Rewrite $9 y^{2}$ as ${(3 y)}^{2}$ .

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

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = 3 y$ .

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

Step 2.4

Multiply $3$ by $- 1$ .

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

Step 3

Factor by grouping.

Tap for more steps...

Step 3.1

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 $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 4$ .

Tap for more steps...

Step 3.1.1

Factor $4$ out of $4 y$ .

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

Step 3.1.2

Rewrite $4$ as $- 2$ plus $6$

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

Step 3.1.3

Apply the distributive property.

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

Step 3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1

Group the first two terms and the last two terms.

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

Step 3.2.2

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

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

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 y - 2$ .

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

Step 4

Cancel the common factor of $2 - 3 y$ and $3 y - 2$ .

Tap for more steps...

Step 4.1

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{(2 + 3 y) (- 1 (- 2) - 3 y)}{(3 y - 2) (y + 2)} \cdot \frac{y^{2} + y - 3}{3 y^{2} + 5 y + 3}$

Step 4.2

Factor $- 1$ out of $- 3 y$ .

$\frac{(2 + 3 y) (- 1 (- 2) - (3 y))}{(3 y - 2) (y + 2)} \cdot \frac{y^{2} + y - 3}{3 y^{2} + 5 y + 3}$

Step 4.3

Factor $- 1$ out of $- 1 (- 2) - (3 y)$ .

$\frac{(2 + 3 y) (- 1 (- 2 + 3 y))}{(3 y - 2) (y + 2)} \cdot \frac{y^{2} + y - 3}{3 y^{2} + 5 y + 3}$

Step 4.4

Reorder terms.

$\frac{(2 + 3 y) (- 1 (3 y - 2))}{(3 y - 2) (y + 2)} \cdot \frac{y^{2} + y - 3}{3 y^{2} + 5 y + 3}$

Step 4.5

Cancel the common factor.

$\frac{(2 + 3 y) (- 1 (3 y - 2))}{(3 y - 2) (y + 2)} \cdot \frac{y^{2} + y - 3}{3 y^{2} + 5 y + 3}$

Step 4.6

Rewrite the expression.

$\frac{(2 + 3 y) \cdot (- 1)}{y + 2} \cdot \frac{y^{2} + y - 3}{3 y^{2} + 5 y + 3}$

Step 5

Simplify the expression.

Tap for more steps...

Step 5.1

Move $- 1$ to the left of $2 + 3 y$ .

$\frac{- 1 \cdot (2 + 3 y)}{y + 2} \cdot \frac{y^{2} + y - 3}{3 y^{2} + 5 y + 3}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{2 + 3 y}{y + 2} \cdot \frac{y^{2} + y - 3}{3 y^{2} + 5 y + 3}$

Step 6

Multiply $\frac{y^{2} + y - 3}{3 y^{2} + 5 y + 3}$ by $\frac{2 + 3 y}{y + 2}$ .

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