Algebra Examples

$\frac{x^{2} - 36}{x^{2} - 12 x + 36}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{x^{2} - 6^{2}}{x^{2} - 12 x + 36}$

Step 1.2

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 $b = 6$ .

$\frac{(x + 6) (x - 6)}{x^{2} - 12 x + 36}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{(x + 6) (x - 6)}{x^{2} - 12 x + 6^{2}}$

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.

$12 x = 2 \cdot x \cdot 6$

Step 2.3

Rewrite the polynomial.

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

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 6$ .

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

Step 3

Cancel the common factor of $x - 6$ and ${(x - 6)}^{2}$ .

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Step 3.1

Factor $x - 6$ out of $(x + 6) (x - 6)$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x - 6$ out of ${(x - 6)}^{2}$ .

$\frac{(x - 6) (x + 6)}{(x - 6) (x - 6)}$

Step 3.2.2

Cancel the common factor.

$\frac{(x - 6) (x + 6)}{(x - 6) (x - 6)}$

Step 3.2.3

Rewrite the expression.

$\frac{x + 6}{x - 6}$