Given:
[tex]\dfrac{5}{6+2i}[/tex]
To find:
The quotient in the form a + bi.
Solution:
We have,
[tex]\dfrac{5}{6+2i}[/tex]
Multiply numerator and denominator by 6-2i.
[tex]=\dfrac{5\times (6-2i)}{(6+2i)\times (6-2i)}[/tex]
[tex]=\dfrac{5(6)+5(-2i)}{6^2+(2i)^2}[/tex]
[tex]=\dfrac{30-10i}{36+4i^2}[/tex]
[tex]=\dfrac{30-10i}{36+4(-1)}[/tex] [tex][\because i^2=-1][/tex]
On further simplification, we get
[tex]=\dfrac{30-10i}{36-4}[/tex]
[tex]=\dfrac{30-10i}{32}[/tex]
[tex]=\dfrac{30}{32}-\dfrac{10i}{32}[/tex]
[tex]=\dfrac{15}{16}+(-\dfrac{5}{16})i[/tex]
Therefore, the form a+bi is [tex]\dfrac{15}{16}+(-\dfrac{5}{16})i[/tex], where, [tex]a=\dfrac{15}{16}[/tex] and [tex]b=(-\dfrac{5}{16})[/tex].
