Precalculus Examples

$\frac{2}{x^{2} - 36} - \frac{1}{x^{2} - 6 x}$

Step 1

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

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Step 1.1

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

$\frac{2}{x^{2} - 36} - \frac{1}{x \cdot x - 6 x}$

Step 1.2

Factor $x$ out of $- 6 x$ .

$\frac{2}{x^{2} - 36} - \frac{1}{x \cdot x + x \cdot - 6}$

Step 1.3

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

$\frac{2}{x^{2} - 36} - \frac{1}{x (x - 6)}$

Step 2

To write $\frac{2}{x^{2} - 36}$ as a fraction with a common denominator, multiply by $\frac{x (x - 6)}{x (x - 6)}$ .

$\frac{2}{x^{2} - 36} \cdot \frac{x (x - 6)}{x (x - 6)} - \frac{1}{x (x - 6)}$

Step 3

To write $- \frac{1}{x (x - 6)}$ as a fraction with a common denominator, multiply by $\frac{x^{2} - 36}{x^{2} - 36}$ .

$\frac{2}{x^{2} - 36} \cdot \frac{x (x - 6)}{x (x - 6)} - \frac{1}{x (x - 6)} \cdot \frac{x^{2} - 36}{x^{2} - 36}$

Step 4

Write each expression with a common denominator of $(x^{2} - 36) x (x - 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{x^{2} - 36}$ by $\frac{x (x - 6)}{x (x - 6)}$ .

$\frac{2 (x (x - 6))}{(x^{2} - 36) (x (x - 6))} - \frac{1}{x (x - 6)} \cdot \frac{x^{2} - 36}{x^{2} - 36}$

Step 4.2

Multiply $\frac{1}{x (x - 6)}$ by $\frac{x^{2} - 36}{x^{2} - 36}$ .

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

Step 4.3

Reorder the factors of $(x^{2} - 36) (x (x - 6))$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Rewrite $\frac{2 (x (x - 6)) - (x^{2} - 36)}{x (x - 6) (x^{2} - 36)}$ in a factored form.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 6.3

Move $- 6$ to the left of $x$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $- 6$ by $2$ .

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

Step 6.6

Apply the distributive property.

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

Step 6.7

Multiply $- 1$ by $- 36$ .

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

Step 6.8

Subtract $x^{2}$ from $2 x^{2}$ .

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

Step 6.9

Factor using the perfect square rule.

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Step 6.9.1

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

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

Step 6.9.2

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 6.9.3

Rewrite the polynomial.

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

Step 6.9.4

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

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

Step 6.10

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

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

Step 6.11

Factor.

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Step 6.11.1

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 $b = 6$ .

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

Step 6.11.2

Remove unnecessary parentheses.

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

Step 6.12

Combine exponents.

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Step 6.12.1

Raise $x - 6$ to the power of $1$ .

$\frac{{(x - 6)}^{2}}{x ({(x - 6)}^{1} (x - 6)) (x + 6)}$

Step 6.12.2

Raise $x - 6$ to the power of $1$ .

$\frac{{(x - 6)}^{2}}{x ({(x - 6)}^{1} {(x - 6)}^{1}) (x + 6)}$

Step 6.12.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(x - 6)}^{2}}{x {(x - 6)}^{1 + 1} (x + 6)}$

Step 6.12.4

Add $1$ and $1$ .

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

Step 6.13

Reduce the expression $\frac{{(x - 6)}^{2}}{x {(x - 6)}^{2} (x + 6)}$ by cancelling the common factors.

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Step 6.13.1

Cancel the common factor.

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

Step 6.13.2

Rewrite the expression.

$\frac{1}{x (x + 6)}$