Question

Find the quotient. Write your answer in standard form.

3 + i / 3 - i

A. -1
B. 1 - i
C. 3/5 + 4/5 i
D. 4/5 + 3/5 i

Answer 1

multiply top and bottom both by the conjugate of the denominator (3-i)

Answer 2

do you know the definition of conjugate ?

Answer 3

no. ): @Nnesha

Answer 4

conjugate example just change the sign of imaginary number
a+bi conjugate is a-bi
bi imaginary
a= real
so 3-i conjugate is what ?

Answer 5

3+i ?? @Nnesha

Answer 6

is it
\[\frac{3+i}{3-i}\]?

Answer 7

if so, multiply top and bottom by the conjugate of the denominator
the conjugate of \(a+bi\) is \(a-bi\) and this works because
\[(a+bi)(a-bi)=a^2+b^2\] a real number

Answer 8

first step is \[\frac{3+i}{3-1}\times \frac{3+i}{3+1}\] which gives you
\[\frac{(3+i)(3+i)}{3^2+1^2}\]

Answer 9

the denominator is evidently \(10\) and the numerator is whatever you get when you multiply that out

Answer 10

you good from there?

Answer 11

sorry typo there
i meant "first step is \[\frac{3+i}{3-i}\times \frac{3+i}{3+i}\]

Answer 12

I got (9 + 6i + i^2) / (9 - i) I don't know where to go from there? @satellite73

Answer 13

ok lets back up, your denominator is wrong

Answer 14

Is it 9-i^2?

Answer 15

it is not \(9-i\) but rather \(9+1=10\)
\[3-i\] has real part \(3\) and imaginary part \(-1\)
when you multiply
\[(3+i)(3-i)\] you get
\[3^2+(-1)^2\] or \(10\)

Answer 16

you can write it that way if you want, but don't forget \(i^2=-1\) so \(9-i^2=9+1=10\)

Answer 17

So is i by itself 1?

Answer 18

you should really just be thinking that
\[(a+bi)(a-bi)=a^2+b^2\] just like in pythagoras

Answer 19

no \(i\neq 1\) but \(i^2=-1\) and so \(-i^2=+1\)

Answer 20

probably the confusing part is how to interpret \(a+bi\) when you have \(3-i\)
in that case \(a=3,b=-1\)
of course when you square, you can ignore the minus sign

Answer 21

your numerator of
\[9+6i+i^2\]is correct, but not complete since \(i^2=-1\) then
\[9+6i+i^2=9+6i-1=8+6i\]

Answer 22

giving you an almost final answer of
\[\frac{8+6i}{10}\] now cancel the common factor of 2 top and bottom

Answer 23

So it's D!! I UNDERSTAND!! Thank you so much. (:

Answer 24

yes it is D, and you are welcome