# How do you write the following expression in standard form (2i)/(2+i)+5/(2-i)?

Oct 23, 2016

Combine the two fractions using a common denominator and then simplify. $\frac{12}{5} + \frac{9}{5} i$

#### Explanation:

Given:
$\frac{2 i}{2 + i} + \frac{5}{2 - i} =$

Make the common denominator $\left(2 + i\right) \left(2 - i\right)$

$\frac{2 i}{2 + i} \left(\frac{2 - i}{2 - i}\right) + \frac{5}{2 - i} \left(\frac{2 + i}{2 + i}\right) =$

Combine over the denominator:

$\frac{\left(2 i\right) \left(2 - i\right) + 5 \left(2 + i\right)}{\left(2 - i\right) \left(2 + i\right)} =$

Multiply the denominator, using the pattern $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$:

$\frac{\left(2 i\right) \left(2 - i\right) + 5 \left(2 + i\right)}{4 - {i}^{2}} =$

Substitute + 1 for $- {i}^{2}$

$\frac{\left(2 i\right) \left(2 - i\right) + 5 \left(2 + i\right)}{4 + 1} =$

$\frac{\left(2 i\right) \left(2 - i\right) + 5 \left(2 + i\right)}{5} =$

Perform the multiplications in the numerator:

$\frac{4 i - 2 {i}^{2} + 10 + 5 i}{5} =$

Replace $- 2 {i}^{2}$ we +2:

$\frac{4 i + 2 + 10 + 5 i}{5} =$

Combine like terms:

$\frac{12 + 9 i}{5} =$

$\frac{12}{5} + \frac{9}{5} i$

Standard form $a + b i$