# Complex Fraction

How to solve a complex fraction: formula, 2 examples, and their solutions.

## Formula

### Formula

A complex fraction is a fraction
whose numerator or denominator is also a fraction
(or both are also fractions).

To solve a complex fraction [a/b]/[c/d],

write the product of the outer factors,
in the numerator

and write the product of the inner factors,
bc,
in the denominator:

## Example 1

### Solution

Write the product of the outer factors,
(x + 2)(x - 1),
in the numerator.

And write the product of the inner factors,
x⋅6 = 6x,
in the denominator.

So (x + 2)(x - 1)/6x is the answer.

## Example 2

### Solution

To solve the given complex fraction,
first combine the numerator part 4x - 1/x.

4x = 4x2/x

4x2/x - 1/x
= [4x2 - 1]/x

4x2 - 1
= (2x)2 - 12

(2x)2 - 12 = (2x + 1)(2x - 1)

Factor the Difference of Two Squares: a2 - b2

4x - 1/x
= (2x + 1)(2x - 1)/x

So (given) = [(2x + 1)(2x - 1)/x] / [(2x - 1)2/1].

To use the complex fraction formula,
change (2x - 1)2 to (2x - 1)2/1.

Cancel the common factor (2x - 1)
in both of
the main numerator and the main denominator.

(2x + 1)(2x - 1)/x → (2x + 1)/x

(2x - 1)2/1 → (2x - 1)/1

Solve the complex fraction.

Write the product of the outer factors,
(2x + 1)⋅1,
in the numerator.

And write the product of the inner factors,
x(2x - 1),
in the denominator.

(2x + 1)⋅1 = 2x + 1

So (2x + 1)/[x(2x - 1)] is the answer.