Algebra Examples

$\frac{8}{x^{2} - 9 x} - \frac{1}{x^{2} - 81}$

Step 1

Simplify each term.

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Step 1.1

Factor $x$ out of $x^{2} - 9 x$ .

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Step 1.1.1

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

$\frac{8}{x \cdot x - 9 x} - \frac{1}{x^{2} - 81}$

Step 1.1.2

Factor $x$ out of $- 9 x$ .

$\frac{8}{x \cdot x + x \cdot - 9} - \frac{1}{x^{2} - 81}$

Step 1.1.3

Factor $x$ out of $x \cdot x + x \cdot - 9$ .

$\frac{8}{x (x - 9)} - \frac{1}{x^{2} - 81}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $81$ as $9^{2}$ .

$\frac{8}{x (x - 9)} - \frac{1}{x^{2} - 9^{2}}$

Step 1.2.2

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 $b = 9$ .

$\frac{8}{x (x - 9)} - \frac{1}{(x + 9) (x - 9)}$

Step 2

To write $\frac{8}{x (x - 9)}$ as a fraction with a common denominator, multiply by $\frac{x + 9}{x + 9}$ .

$\frac{8}{x (x - 9)} \cdot \frac{x + 9}{x + 9} - \frac{1}{(x + 9) (x - 9)}$

Step 3

To write $- \frac{1}{(x + 9) (x - 9)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{8}{x (x - 9)} \cdot \frac{x + 9}{x + 9} - \frac{1}{(x + 9) (x - 9)} \cdot \frac{x}{x}$

Step 4

Write each expression with a common denominator of $x (x - 9) (x + 9)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{8}{x (x - 9)}$ by $\frac{x + 9}{x + 9}$ .

$\frac{8 (x + 9)}{x (x - 9) (x + 9)} - \frac{1}{(x + 9) (x - 9)} \cdot \frac{x}{x}$

Step 4.2

Multiply $\frac{1}{(x + 9) (x - 9)}$ by $\frac{x}{x}$ .

$\frac{8 (x + 9)}{x (x - 9) (x + 9)} - \frac{x}{(x + 9) (x - 9) x}$

Step 4.3

Reorder the factors of $x (x - 9) (x + 9)$ .

$\frac{8 (x + 9)}{x (x + 9) (x - 9)} - \frac{x}{(x + 9) (x - 9) x}$

Step 4.4

Reorder the factors of $(x + 9) (x - 9) x$ .

$\frac{8 (x + 9)}{x (x + 9) (x - 9)} - \frac{x}{x (x + 9) (x - 9)}$

Step 5

Combine the numerators over the common denominator.

$\frac{8 (x + 9) - x}{x (x + 9) (x - 9)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{8 x + 8 \cdot 9 - x}{x (x + 9) (x - 9)}$

Step 6.2

Multiply $8$ by $9$ .

$\frac{8 x + 72 - x}{x (x + 9) (x - 9)}$

Step 6.3

Subtract $x$ from $8 x$ .

$\frac{7 x + 72}{x (x + 9) (x - 9)}$