Basic Math Examples

$\frac{y^{2} - 2 y - 3}{\frac{y^{2} - 3 y - 4}{\frac{y^{2} + 3 y + 2}{y^{2} - 2 y - 8}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(y^{2} - 2 y - 3) \frac{\frac{y^{2} + 3 y + 2}{y^{2} - 2 y - 8}}{y^{2} - 3 y - 4}$

Step 2

Factor $y^{2} + 3 y + 2$ 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 $2$ and whose sum is $3$ .

$1, 2$

Step 2.2

Write the factored form using these integers.

$(y^{2} - 2 y - 3) \frac{\frac{(y + 1) (y + 2)}{y^{2} - 2 y - 8}}{y^{2} - 3 y - 4}$

Step 3

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

$- 4, 2$

Step 3.2

Write the factored form using these integers.

$(y^{2} - 2 y - 3) \frac{\frac{(y + 1) (y + 2)}{(y - 4) (y + 2)}}{y^{2} - 3 y - 4}$

Step 4

Reduce the expression $\frac{(y + 1) (y + 2)}{(y - 4) (y + 2)}$ by cancelling the common factors.

Tap for more steps...

Step 4.1

Cancel the common factor.

$(y^{2} - 2 y - 3) \frac{\frac{(y + 1) (y + 2)}{(y - 4) (y + 2)}}{y^{2} - 3 y - 4}$

Step 4.2

Rewrite the expression.

$(y^{2} - 2 y - 3) \frac{\frac{y + 1}{y - 4}}{y^{2} - 3 y - 4}$

Step 5

Factor $y^{2} - 3 y - 4$ 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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 5.2

Write the factored form using these integers.

$(y^{2} - 2 y - 3) \frac{\frac{y + 1}{y - 4}}{(y - 4) (y + 1)}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$(y^{2} - 2 y - 3) (\frac{y + 1}{y - 4} \cdot \frac{1}{(y - 4) (y + 1)})$

Step 7

Cancel the common factor of $y + 1$ .

Tap for more steps...

Step 7.1

Factor $y + 1$ out of $(y - 4) (y + 1)$ .

$(y^{2} - 2 y - 3) (\frac{y + 1}{y - 4} \cdot \frac{1}{(y + 1) (y - 4)})$

Step 7.2

Cancel the common factor.

$(y^{2} - 2 y - 3) (\frac{y + 1}{y - 4} \cdot \frac{1}{(y + 1) (y - 4)})$

Step 7.3

Rewrite the expression.

$(y^{2} - 2 y - 3) (\frac{1}{y - 4} \cdot \frac{1}{y - 4})$

Step 8

Multiply $\frac{1}{y - 4}$ by $\frac{1}{y - 4}$ .

$(y^{2} - 2 y - 3) \frac{1}{(y - 4) (y - 4)}$

Step 9

Raise $y - 4$ to the power of $1$ .

$(y^{2} - 2 y - 3) \frac{1}{{(y - 4)}^{1} (y - 4)}$

Step 10

Raise $y - 4$ to the power of $1$ .

$(y^{2} - 2 y - 3) \frac{1}{{(y - 4)}^{1} {(y - 4)}^{1}}$

Step 11

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

$(y^{2} - 2 y - 3) \frac{1}{{(y - 4)}^{1 + 1}}$

Step 12

Combine fractions.

Tap for more steps...

Step 12.1

Add $1$ and $1$ .

$(y^{2} - 2 y - 3) \frac{1}{{(y - 4)}^{2}}$

Step 12.2

Multiply $y^{2} - 2 y - 3$ by $\frac{1}{{(y - 4)}^{2}}$ .

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

Step 13

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

Tap for more steps...

Step 13.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 13.2

Write the factored form using these integers.

$\frac{(y - 3) (y + 1)}{{(y - 4)}^{2}}$