Question

How do you express `1/(2+i)` in `a+bi` form?

Answer 1

`1/(2+i)=2/5-1/5i`

Explanation:

In order to transform the complex number `1"/"(alpha+betai)` into the form `a+bi`, we multiply the original number by its complex conjugate.

`1/(2+i)=(2-i)/((2+i)(2-i))=(2-i)/(2^2-i^2)=(2-i)/(4+1)=(2-i)/5=2/5-1/5i`

Answer 2

`2/5-1/5i`

Explanation:

`"multiply the numerator/denominator by the"`

`color(blue)"complex conjugate ""of the denominator"`

`"the conjugate of "2+i" is "2color(red)(-)i`

`rArr((2-i))/((2+i)(2-i))`

`=(2-i)/(4+2i-2i-i^2)`

`[i^2=(sqrt(-1))^2=-1]`

`=(2-i)/5=2/5-1/5ilarrcolor(red)"in standard form"`