Simplify (4/(x^2)-1)/(2/x+4/(x^2))

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

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

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

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

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

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

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

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

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 $b = x$ .

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

Step 5

Simplify terms.

Factor $2$ out of $x \cdot 2 + 4$ .

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

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

Factor $2$ out of $4$ .

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

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

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

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (4/(x^2)-1)/(2/x+4/(x^2))

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