Basic Math Examples

$\frac{p^{2} - 2 p - 15}{p^{2} - 25} \cdot \frac{p^{2} + 4 p - 5}{5 p - 5}$

Step 1

Factor $p^{2} - 2 p - 15$ using the AC method.

Tap for more steps...

Step 1.1

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 $- 2$ .

$- 5, 3$

Step 1.2

Write the factored form using these integers.

$\frac{(p - 5) (p + 3)}{p^{2} - 25} \cdot \frac{p^{2} + 4 p - 5}{5 p - 5}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

Rewrite $25$ as $5^{2}$ .

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = p$ and $b = 5$ .

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

Step 3

Factor $p^{2} + 4 p - 5$ 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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 3.2

Write the factored form using these integers.

$\frac{(p - 5) (p + 3)}{(p + 5) (p - 5)} \cdot \frac{(p - 1) (p + 5)}{5 p - 5}$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Factor $5$ out of $5 p - 5$ .

Tap for more steps...

Step 4.1.1

Factor $5$ out of $5 p$ .

$\frac{(p - 5) (p + 3)}{(p + 5) (p - 5)} \cdot \frac{(p - 1) (p + 5)}{5 (p) - 5}$

Step 4.1.2

Factor $5$ out of $- 5$ .

$\frac{(p - 5) (p + 3)}{(p + 5) (p - 5)} \cdot \frac{(p - 1) (p + 5)}{5 (p) + 5 (- 1)}$

Step 4.1.3

Factor $5$ out of $5 (p) + 5 (- 1)$ .

$\frac{(p - 5) (p + 3)}{(p + 5) (p - 5)} \cdot \frac{(p - 1) (p + 5)}{5 (p - 1)}$

Step 4.2

Cancel the common factor of $p + 5$ .

Tap for more steps...

Step 4.2.1

Factor $p + 5$ out of $(p - 1) (p + 5)$ .

$\frac{(p - 5) (p + 3)}{(p + 5) (p - 5)} \cdot \frac{(p + 5) (p - 1)}{5 (p - 1)}$

Step 4.2.2

Cancel the common factor.

$\frac{(p - 5) (p + 3)}{(p + 5) (p - 5)} \cdot \frac{(p + 5) (p - 1)}{5 (p - 1)}$

Step 4.2.3

Rewrite the expression.

$\frac{(p - 5) (p + 3)}{p - 5} \cdot \frac{p - 1}{5 (p - 1)}$

Step 4.3

Multiply $\frac{(p - 5) (p + 3)}{p - 5}$ by $\frac{p - 1}{5 (p - 1)}$ .

$\frac{(p - 5) (p + 3) (p - 1)}{(p - 5) (5 (p - 1))}$

Step 4.4

Cancel the common factor of $p - 5$ .

Tap for more steps...

Step 4.4.1

Cancel the common factor.

$\frac{(p - 5) (p + 3) (p - 1)}{(p - 5) (5 (p - 1))}$

Step 4.4.2

Rewrite the expression.

$\frac{(p + 3) (p - 1)}{5 (p - 1)}$

Step 4.5

Cancel the common factor of $p - 1$ .

Tap for more steps...

Step 4.5.1

Cancel the common factor.

$\frac{(p + 3) (p - 1)}{5 (p - 1)}$

Step 4.5.2

Rewrite the expression.

$\frac{p + 3}{5}$