Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{x}{x^{2} - 6^{2}} + \frac{4}{x}$

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 $b = 6$ .

$\frac{x}{(x + 6) (x - 6)} + \frac{4}{x}$

Step 2

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

$\frac{x}{(x + 6) (x - 6)} \cdot \frac{x}{x} + \frac{4}{x}$

Step 3

To write $\frac{4}{x}$ as a fraction with a common denominator, multiply by $\frac{(x + 6) (x - 6)}{(x + 6) (x - 6)}$ .

$\frac{x}{(x + 6) (x - 6)} \cdot \frac{x}{x} + \frac{4}{x} \cdot \frac{(x + 6) (x - 6)}{(x + 6) (x - 6)}$

Step 4

Write each expression with a common denominator of $(x + 6) (x - 6) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x}{(x + 6) (x - 6)}$ by $\frac{x}{x}$ .

$\frac{x \cdot x}{(x + 6) (x - 6) x} + \frac{4}{x} \cdot \frac{(x + 6) (x - 6)}{(x + 6) (x - 6)}$

Step 4.2

Multiply $\frac{4}{x}$ by $\frac{(x + 6) (x - 6)}{(x + 6) (x - 6)}$ .

$\frac{x \cdot x}{(x + 6) (x - 6) x} + \frac{4 ((x + 6) (x - 6))}{x ((x + 6) (x - 6))}$

Step 4.3

Reorder the factors of $(x + 6) (x - 6) x$ .

$\frac{x \cdot x}{x (x + 6) (x - 6)} + \frac{4 ((x + 6) (x - 6))}{x ((x + 6) (x - 6))}$

Step 4.4

Reorder the factors of $x ((x + 6) (x - 6))$ .

$\frac{x \cdot x}{x (x + 6) (x - 6)} + \frac{4 ((x + 6) (x - 6))}{x (x + 6) (x - 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{x \cdot x + 4 ((x + 6) (x - 6))}{x (x + 6) (x - 6)}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $x$ by $x$ .

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

Step 6.2

Apply the distributive property.

$\frac{x^{2} + (4 x + 4 \cdot 6) (x - 6)}{x (x + 6) (x - 6)}$

Step 6.3

Multiply $4$ by $6$ .

$\frac{x^{2} + (4 x + 24) (x - 6)}{x (x + 6) (x - 6)}$

Step 6.4

Expand $(4 x + 24) (x - 6)$ using the FOIL Method.

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Step 6.4.1

Apply the distributive property.

$\frac{x^{2} + 4 x (x - 6) + 24 (x - 6)}{x (x + 6) (x - 6)}$

Step 6.4.2

Apply the distributive property.

$\frac{x^{2} + 4 x \cdot x + 4 x \cdot - 6 + 24 (x - 6)}{x (x + 6) (x - 6)}$

Step 6.4.3

Apply the distributive property.

$\frac{x^{2} + 4 x \cdot x + 4 x \cdot - 6 + 24 x + 24 \cdot - 6}{x (x + 6) (x - 6)}$

Step 6.5

Simplify and combine like terms.

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Step 6.5.1

Simplify each term.

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Step 6.5.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 6.5.1.1.1

Move $x$ .

$\frac{x^{2} + 4 (x \cdot x) + 4 x \cdot - 6 + 24 x + 24 \cdot - 6}{x (x + 6) (x - 6)}$

Step 6.5.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 4 x^{2} + 4 x \cdot - 6 + 24 x + 24 \cdot - 6}{x (x + 6) (x - 6)}$

Step 6.5.1.2

Multiply $- 6$ by $4$ .

$\frac{x^{2} + 4 x^{2} - 24 x + 24 x + 24 \cdot - 6}{x (x + 6) (x - 6)}$

Step 6.5.1.3

Multiply $24$ by $- 6$ .

$\frac{x^{2} + 4 x^{2} - 24 x + 24 x - 144}{x (x + 6) (x - 6)}$

Step 6.5.2

Add $- 24 x$ and $24 x$ .

$\frac{x^{2} + 4 x^{2} + 0 - 144}{x (x + 6) (x - 6)}$

Step 6.5.3

Add $4 x^{2}$ and $0$ .

$\frac{x^{2} + 4 x^{2} - 144}{x (x + 6) (x - 6)}$

Step 6.6

Add $x^{2}$ and $4 x^{2}$ .

$\frac{5 x^{2} - 144}{x (x + 6) (x - 6)}$