Pre-Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $6$ and $2$ .

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Step 1.1.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 3}{2 x^{2}} + \frac{x}{x^{2} - x}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

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

$\frac{2 \cdot 3}{2 (x^{2})} + \frac{x}{x^{2} - x}$

Step 1.1.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 x^{2}} + \frac{x}{x^{2} - x}$

Step 1.1.2.3

Rewrite the expression.

$\frac{3}{x^{2}} + \frac{x}{x^{2} - x}$

Step 1.2

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

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Step 1.2.1

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

$\frac{3}{x^{2}} + \frac{x}{x \cdot x - x}$

Step 1.2.2

Factor $x$ out of $- x$ .

$\frac{3}{x^{2}} + \frac{x}{x \cdot x + x \cdot - 1}$

Step 1.2.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$\frac{3}{x^{2}} + \frac{x}{x (x - 1)}$

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

$\frac{3}{x^{2}} + \frac{x}{x (x - 1)}$

Step 1.3.2

Rewrite the expression.

$\frac{3}{x^{2}} + \frac{1}{x - 1}$

Step 2

To write $\frac{3}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{3}{x^{2}} \cdot \frac{x - 1}{x - 1} + \frac{1}{x - 1}$

Step 3

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

$\frac{3}{x^{2}} \cdot \frac{x - 1}{x - 1} + \frac{1}{x - 1} \cdot \frac{x^{2}}{x^{2}}$

Step 4

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

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Step 4.1

Multiply $\frac{3}{x^{2}}$ by $\frac{x - 1}{x - 1}$ .

$\frac{3 (x - 1)}{x^{2} (x - 1)} + \frac{1}{x - 1} \cdot \frac{x^{2}}{x^{2}}$

Step 4.2

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

$\frac{3 (x - 1)}{x^{2} (x - 1)} + \frac{x^{2}}{(x - 1) x^{2}}$

Step 4.3

Reorder the factors of $(x - 1) x^{2}$ .

$\frac{3 (x - 1)}{x^{2} (x - 1)} + \frac{x^{2}}{x^{2} (x - 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{3 (x - 1) + x^{2}}{x^{2} (x - 1)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{3 x + 3 \cdot - 1 + x^{2}}{x^{2} (x - 1)}$

Step 6.2

Multiply $3$ by $- 1$ .

$\frac{3 x - 3 + x^{2}}{x^{2} (x - 1)}$

Step 6.3

Reorder terms.

$\frac{x^{2} + 3 x - 3}{x^{2} (x - 1)}$