Simplify (a-2)/(4/a-a)

$\frac{a - 2}{\frac{4}{a} - a}$

Step 1

Multiply the numerator and denominator of the complex fraction by $a$ .

Multiply $\frac{a - 2}{\frac{4}{a} - a}$ by $\frac{a}{a}$ .

$\frac{a}{a} \cdot \frac{a - 2}{\frac{4}{a} - a}$

Combine.

$\frac{a (a - 2)}{a (\frac{4}{a} - a)}$

Step 2

Apply the distributive property.

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

Step 3

Cancel the common factor of $a$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

Multiply $a$ by $a$ .

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

Move $- 2$ to the left of $a$ .

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

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

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

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

Factor $a$ out of $- 2 a$ .

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

Factor $a$ out of $a \cdot a + a \cdot - 2$ .

$\frac{a (a - 2)}{4 + a (- a)}$

Step 5

Simplify the denominator.

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

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

Rewrite $a (a)$ as $a^{2}$ .

$\frac{a (a - 2)}{2^{2} - a^{2}}$

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 $b = a$ .

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

Step 6

Reduce the expression by cancelling the common factors.

Cancel the common factor of $a - 2$ and $2 - a$ .

Factor $- 1$ out of $a$ .

$\frac{a (- 1 (- a) - 2)}{(2 + a) (2 - a)}$

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{a (- 1 (- a) - 1 (2))}{(2 + a) (2 - a)}$

Factor $- 1$ out of $- 1 (- a) - 1 (2)$ .

$\frac{a (- 1 (- a + 2))}{(2 + a) (2 - a)}$

Reorder terms.

$\frac{a (- 1 (- a + 2))}{(2 + a) (- a + 2)}$

Cancel the common factor.

$\frac{a (- 1 (- a + 2))}{(2 + a) (- a + 2)}$

Rewrite the expression.

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

Simplify the expression.

Move $- 1$ to the left of $a$ .

$\frac{- 1 \cdot a}{2 + a}$

Move the negative in front of the fraction.

$- \frac{a}{2 + a}$

Simplify (a-2)/(4/a-a)

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