Pre-Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

$\frac{x}{x \cdot x + 4 x} + \frac{2}{x^{2}}$

Step 1.1.2

Factor $x$ out of $4 x$ .

$\frac{x}{x \cdot x + x \cdot 4} + \frac{2}{x^{2}}$

Step 1.1.3

Factor $x$ out of $x \cdot x + x \cdot 4$ .

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

$\frac{1}{x + 4} + \frac{2}{x^{2}}$

Step 2

To write $\frac{1}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{1}{x + 4} \cdot \frac{x^{2}}{x^{2}} + \frac{2}{x^{2}}$

Step 3

To write $\frac{2}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{1}{x + 4} \cdot \frac{x^{2}}{x^{2}} + \frac{2}{x^{2}} \cdot \frac{x + 4}{x + 4}$

Step 4

Write each expression with a common denominator of $(x + 4) x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{x + 4}$ by $\frac{x^{2}}{x^{2}}$ .

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

Step 4.2

Multiply $\frac{2}{x^{2}}$ by $\frac{x + 4}{x + 4}$ .

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

Step 4.3

Reorder the factors of $(x + 4) x^{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $2$ by $4$ .

$\frac{x^{2} + 2 x + 8}{x^{2} (x + 4)}$