Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $2 x^{2}$ .

Tap for more steps...

Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Tap for more steps...

Step 3.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $2$ by $2$ .

$\frac{4 + 2 x^{2} (- \frac{1}{2})}{2 x^{2} \frac{x^{2}}{2} + 2 x^{2} x}$

Step 3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 3.3.2

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

$\frac{4 + 2 (x^{2}) \frac{- 1}{2}}{2 x^{2} \frac{x^{2}}{2} + 2 x^{2} x}$

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 3.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.5.1

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

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

Step 3.5.2

Add $2$ and $2$ .

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

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Move $- 1$ to the left of $x^{2}$ .

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

Step 4.2

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Step 4.3

Rewrite $4 - x^{2}$ in a factored form.

Tap for more steps...

Step 4.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x$ .

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

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (2 x)$ .

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

Factor $x$ out of $2 x$ .

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

Step 5.2.3

Factor $x$ out of $x \cdot x + x \cdot 2$ .

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

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

Step 5.3

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 5.3.1

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 5.3.1.1

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

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

Step 5.3.1.2

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

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

Step 5.3.2

Add $2$ and $1$ .

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

Step 6

Cancel the common factor of $2 + x$ and $x + 2$ .

Tap for more steps...

Step 6.1

Reorder terms.

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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