Basic Math Examples

$\frac{x^{2} - 6 x - 27}{x^{2} - 6 x + 9}$

Step 1

Factor $x^{2} - 6 x - 27$ using the AC method.

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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 $- 27$ and whose sum is $- 6$ .

$- 9, 3$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 9) (x + 3)}{x^{2} - 6 x + 9}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$\frac{(x - 9) (x + 3)}{x^{2} - 6 x + 3^{2}}$

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.

$6 x = 2 \cdot x \cdot 3$

Step 2.3

Rewrite the polynomial.

$\frac{(x - 9) (x + 3)}{x^{2} - 2 \cdot x \cdot 3 + 3^{2}}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{(x - 9) (x + 3)}{{(x - 3)}^{2}}$