Trigonometry Examples

$\frac{\frac{4 y^{2} - 9}{2 y^{2} + 9 y - 18}}{\frac{2 y^{2} + y - 3}{y^{2} + 5 y - 6}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 y^{2} - 9}{2 y^{2} + 9 y - 18} \cdot \frac{y^{2} + 5 y - 6}{2 y^{2} + y - 3}$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$\frac{{(2 y)}^{2} - 9}{2 y^{2} + 9 y - 18} \cdot \frac{y^{2} + 5 y - 6}{2 y^{2} + y - 3}$

Step 2.2

Rewrite $9$ as $3^{2}$ .

$\frac{{(2 y)}^{2} - 3^{2}}{2 y^{2} + 9 y - 18} \cdot \frac{y^{2} + 5 y - 6}{2 y^{2} + y - 3}$

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 $a = 2 y$ and $b = 3$ .

$\frac{(2 y + 3) (2 y - 3)}{2 y^{2} + 9 y - 18} \cdot \frac{y^{2} + 5 y - 6}{2 y^{2} + y - 3}$

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 $a \cdot c = 2 \cdot - 18 = - 36$ and whose sum is $b = 9$ .

Tap for more steps...

Step 3.1.1

Factor $9$ out of $9 y$ .

$\frac{(2 y + 3) (2 y - 3)}{2 y^{2} + 9 (y) - 18} \cdot \frac{y^{2} + 5 y - 6}{2 y^{2} + y - 3}$

Step 3.1.2

Rewrite $9$ as $- 3$ plus $12$

$\frac{(2 y + 3) (2 y - 3)}{2 y^{2} + (- 3 + 12) y - 18} \cdot \frac{y^{2} + 5 y - 6}{2 y^{2} + y - 3}$

Step 3.1.3

Apply the distributive property.

$\frac{(2 y + 3) (2 y - 3)}{2 y^{2} - 3 y + 12 y - 18} \cdot \frac{y^{2} + 5 y - 6}{2 y^{2} + y - 3}$

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.

$\frac{(2 y + 3) (2 y - 3)}{(2 y^{2} - 3 y) + 12 y - 18} \cdot \frac{y^{2} + 5 y - 6}{2 y^{2} + y - 3}$

Step 3.2.2

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

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

Step 3.3

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

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

Step 4

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

Tap for more steps...

Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Factor $y^{2} + 5 y - 6$ 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 $- 6$ and whose sum is $5$ .

$- 1, 6$

Step 5.2

Write the factored form using these integers.

$\frac{2 y + 3}{y + 6} \cdot \frac{(y - 1) (y + 6)}{2 y^{2} + y - 3}$

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 1$ .

Tap for more steps...

Step 6.1.1

Multiply by $1$ .

$\frac{2 y + 3}{y + 6} \cdot \frac{(y - 1) (y + 6)}{2 y^{2} + 1 y - 3}$

Step 6.1.2

Rewrite $1$ as $- 2$ plus $3$

$\frac{2 y + 3}{y + 6} \cdot \frac{(y - 1) (y + 6)}{2 y^{2} + (- 2 + 3) y - 3}$

Step 6.1.3

Apply the distributive property.

$\frac{2 y + 3}{y + 6} \cdot \frac{(y - 1) (y + 6)}{2 y^{2} - 2 y + 3 y - 3}$

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.

$\frac{2 y + 3}{y + 6} \cdot \frac{(y - 1) (y + 6)}{(2 y^{2} - 2 y) + 3 y - 3}$

Step 6.2.2

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

$\frac{2 y + 3}{y + 6} \cdot \frac{(y - 1) (y + 6)}{2 y (y - 1) + 3 (y - 1)}$

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $y - 1$ .

$\frac{2 y + 3}{y + 6} \cdot \frac{(y - 1) (y + 6)}{(y - 1) (2 y + 3)}$

Step 7

Cancel the common factor of $2 y + 3$ .

Tap for more steps...

Step 7.1

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

$\frac{2 y + 3}{y + 6} \cdot \frac{(y - 1) (y + 6)}{(2 y + 3) (y - 1)}$

Step 7.2

Cancel the common factor.

$\frac{2 y + 3}{y + 6} \cdot \frac{(y - 1) (y + 6)}{(2 y + 3) (y - 1)}$

Step 7.3

Rewrite the expression.

$\frac{1}{y + 6} \cdot \frac{(y - 1) (y + 6)}{y - 1}$

Step 8

Cancel the common factor of $y + 6$ .

Tap for more steps...

Step 8.1

Factor $y + 6$ out of $(y - 1) (y + 6)$ .

$\frac{1}{y + 6} \cdot \frac{(y + 6) (y - 1)}{y - 1}$

Step 8.2

Cancel the common factor.

$\frac{1}{y + 6} \cdot \frac{(y + 6) (y - 1)}{y - 1}$

Step 8.3

Rewrite the expression.

$\frac{y - 1}{y - 1}$

Step 9

Cancel the common factor of $y - 1$ .

Tap for more steps...

Step 9.1

Cancel the common factor.

$\frac{y - 1}{y - 1}$

Step 9.2

Rewrite the expression.

$1$