Pre-Algebra Examples

$\frac{x^{2} - 2 x - 3}{x^{2} - 4 x + 4}$

Step 1

Factor $x^{2} - 2 x - 3$ 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.

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

Step 2

Factor using the perfect square rule.

Tap for more steps...

Step 2.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 2.3

Rewrite the polynomial.

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

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 3

Expand $(x - 3) (x + 1)$ using the FOIL Method.

Tap for more steps...

Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Multiply $x$ by $x$ .

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

Step 4.1.2

Multiply $x$ by $1$ .

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

Step 4.1.3

Multiply $- 3$ by $1$ .

$\frac{x^{2} + x - 3 x - 3}{{(x - 2)}^{2}}$

Step 4.2

Subtract $3 x$ from $x$ .

$\frac{x^{2} - 2 x - 3}{{(x - 2)}^{2}}$

Step 5

Split the fraction $\frac{x^{2} - 2 x - 3}{{(x - 2)}^{2}}$ into two fractions.

$\frac{x^{2} - 2 x}{{(x - 2)}^{2}} + \frac{- 3}{{(x - 2)}^{2}}$

Step 6

Split the fraction $\frac{x^{2} - 2 x}{{(x - 2)}^{2}}$ into two fractions.

$\frac{x^{2}}{{(x - 2)}^{2}} + \frac{- 2 x}{{(x - 2)}^{2}} + \frac{- 3}{{(x - 2)}^{2}}$

Step 7

Move the negative in front of the fraction.

$\frac{x^{2}}{{(x - 2)}^{2}} - \frac{2 x}{{(x - 2)}^{2}} + \frac{- 3}{{(x - 2)}^{2}}$

Step 8

Move the negative in front of the fraction.

$\frac{x^{2}}{{(x - 2)}^{2}} - \frac{2 x}{{(x - 2)}^{2}} - \frac{3}{{(x - 2)}^{2}}$