Basic Math Examples

$\frac{a^{2} - 2 a - 3}{2 a^{2} - 5 a - 3}$

Step 1

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

$- 3, 1$

Step 1.2

Write the factored form using these integers.

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

Step 2

Factor by grouping.

Tap for more steps...

Step 2.1

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 $b = - 5$ .

Tap for more steps...

Step 2.1.1

Factor $- 5$ out of $- 5 a$ .

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

Step 2.1.2

Rewrite $- 5$ as $1$ plus $- 6$

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

Step 2.1.3

Apply the distributive property.

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

Step 2.1.4

Multiply $a$ by $1$ .

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

Step 2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.2.1

Group the first two terms and the last two terms.

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

Step 2.2.2

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

$\frac{(a - 3) (a + 1)}{a (2 a + 1) - 3 (2 a + 1)}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 a + 1$ .

$\frac{(a - 3) (a + 1)}{(2 a + 1) (a - 3)}$

Step 3

Cancel the common factor of $a - 3$ .

Tap for more steps...

Step 3.1

Cancel the common factor.

$\frac{(a - 3) (a + 1)}{(2 a + 1) (a - 3)}$

Step 3.2

Rewrite the expression.

$\frac{a + 1}{2 a + 1}$