Basic Math Examples

$\frac{\frac{y^{2} + 8 y + 15}{y^{2} + 9 y + 18}}{\frac{y^{2} - 9 y + 14}{y^{2} + 4 y - 12}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2} + 8 y + 15}{y^{2} + 9 y + 18} \cdot \frac{y^{2} + 4 y - 12}{y^{2} - 9 y + 14}$

Step 2

Factor $y^{2} + 8 y + 15$ 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 $15$ and whose sum is $8$ .

$3, 5$

Step 2.2

Write the factored form using these integers.

$\frac{(y + 3) (y + 5)}{y^{2} + 9 y + 18} \cdot \frac{y^{2} + 4 y - 12}{y^{2} - 9 y + 14}$

Step 3

Factor $y^{2} + 9 y + 18$ 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 $18$ and whose sum is $9$ .

$3, 6$

Step 3.2

Write the factored form using these integers.

$\frac{(y + 3) (y + 5)}{(y + 3) (y + 6)} \cdot \frac{y^{2} + 4 y - 12}{y^{2} - 9 y + 14}$

Step 4

Cancel the common factor of $y + 3$ .

Tap for more steps...

Step 4.1

Cancel the common factor.

$\frac{(y + 3) (y + 5)}{(y + 3) (y + 6)} \cdot \frac{y^{2} + 4 y - 12}{y^{2} - 9 y + 14}$

Step 4.2

Rewrite the expression.

$\frac{y + 5}{y + 6} \cdot \frac{y^{2} + 4 y - 12}{y^{2} - 9 y + 14}$

Step 5

Factor $y^{2} + 4 y - 12$ 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 $4$ .

$- 2, 6$

Step 5.2

Write the factored form using these integers.

$\frac{y + 5}{y + 6} \cdot \frac{(y - 2) (y + 6)}{y^{2} - 9 y + 14}$

Step 6

Factor $y^{2} - 9 y + 14$ using the AC method.

Tap for more steps...

Step 6.1

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 6.2

Write the factored form using these integers.

$\frac{y + 5}{y + 6} \cdot \frac{(y - 2) (y + 6)}{(y - 7) (y - 2)}$

Step 7

Simplify terms.

Tap for more steps...

Step 7.1

Cancel the common factor of $y + 6$ .

Tap for more steps...

Step 7.1.1

Factor $y + 6$ out of $(y - 2) (y + 6)$ .

$\frac{y + 5}{y + 6} \cdot \frac{(y + 6) (y - 2)}{(y - 7) (y - 2)}$

Step 7.1.2

Cancel the common factor.

$\frac{y + 5}{y + 6} \cdot \frac{(y + 6) (y - 2)}{(y - 7) (y - 2)}$

Step 7.1.3

Rewrite the expression.

$(y + 5) \frac{y - 2}{(y - 7) (y - 2)}$

Step 7.2

Cancel the common factor of $y - 2$ .

Tap for more steps...

Step 7.2.1

Cancel the common factor.

$(y + 5) \frac{y - 2}{(y - 7) (y - 2)}$

Step 7.2.2

Rewrite the expression.

$(y + 5) \frac{1}{y - 7}$

Step 7.3

Multiply $y + 5$ by $\frac{1}{y - 7}$ .

$\frac{y + 5}{y - 7}$