Precalculus Examples

$\frac{x^{2} - 15 x + 54}{x^{4} - 36 x^{2}}$

Step 1

Factor $x^{2} - 15 x + 54$ 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 $54$ and whose sum is $- 15$ .

$- 9, - 6$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 9) (x - 6)}{x^{4} - 36 x^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $x^{2}$ out of $x^{4} - 36 x^{2}$ .

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Step 2.1.1

Factor $x^{2}$ out of $x^{4}$ .

$\frac{(x - 9) (x - 6)}{x^{2} x^{2} - 36 x^{2}}$

Step 2.1.2

Factor $x^{2}$ out of $- 36 x^{2}$ .

$\frac{(x - 9) (x - 6)}{x^{2} x^{2} + x^{2} \cdot - 36}$

Step 2.1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 36$ .

$\frac{(x - 9) (x - 6)}{x^{2} (x^{2} - 36)}$

Step 2.2

Rewrite $36$ as $6^{2}$ .

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

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

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

Step 3

Cancel the common factor of $x - 6$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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