How do you simplify #(3+2i)/(2+i)#?

1 Answer
Mar 6, 2018

Multiply both the top and bottom by the conjugate of the denominator, #(2-i)#, and simplify to get #8/5-1/5i#

Explanation:

Starting with #(3+2i)/(2+i)#, we can get #i# out of the denominator by multiplying both the numerator and denominator by the "conjugate" of the denominator, which is just the denominator with the sign switched in the middle:
#(3+2i)/(2+i)*(2-i)/(2-i)#
Multiply and simplify (remember that #i=sqrt(-1)# so #i^2=-1#) to get

#(6-3i+4i-2i^2)/(4-i^2)=(6+i-2(-1))/(4-(-1))=(6+i+2)/5=(8+i)/5#

Finally, your teacher may or may not care about this, but "standard form" for a complex number is #a+-bi#, where #a# is the Real number part. Put your answer in this format by breaking up the numerator:
#(8+i)/5=8/5-1/5i#