Precalculus Examples

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

Step 1

Factor using the perfect square rule.

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Step 1.1

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

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

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.

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

Step 1.3

Rewrite the polynomial.

$\frac{x^{2} - 2 \cdot x \cdot 3 + 3^{2}}{x^{2} - x - 6}$

Step 1.4

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

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

Step 2

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

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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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 2.2

Write the factored form using these integers.

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

Step 3

Cancel the common factor of ${(x - 3)}^{2}$ and $x - 3$ .

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Step 3.1

Factor $x - 3$ out of ${(x - 3)}^{2}$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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