Precalculus Examples

$\frac{x^{2} - 4 x + 4}{x^{2} - 7 x + 10}$

Step 1

Factor using the perfect square rule.

Tap for more steps...

Step 1.1

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

$\frac{x^{2} - 4 x + 2^{2}}{x^{2} - 7 x + 10}$

Step 1.2

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 1.3

Rewrite the polynomial.

$\frac{x^{2} - 2 \cdot x \cdot 2 + 2^{2}}{x^{2} - 7 x + 10}$

Step 1.4

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

$\frac{{(x - 2)}^{2}}{x^{2} - 7 x + 10}$

Step 2

Factor $x^{2} - 7 x + 10$ 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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Step 2.2

Write the factored form using these integers.

$\frac{{(x - 2)}^{2}}{(x - 5) (x - 2)}$

Step 3

Cancel the common factor of ${(x - 2)}^{2}$ and $x - 2$ .

Tap for more steps...

Step 3.1

Factor $x - 2$ out of ${(x - 2)}^{2}$ .

$\frac{(x - 2) (x - 2)}{(x - 5) (x - 2)}$

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Factor $x - 2$ out of $(x - 5) (x - 2)$ .

$\frac{(x - 2) (x - 2)}{(x - 2) (x - 5)}$

Step 3.2.2

Cancel the common factor.

$\frac{(x - 2) (x - 2)}{(x - 2) (x - 5)}$

Step 3.2.3

Rewrite the expression.

$\frac{x - 2}{x - 5}$