Find the quotient. Express your answer in standard form. 6+i4−2i\frac{6+i}{4-2i}

1110+45i\frac{11}{10}+\frac45i1011+54i

Find the quotient. Express your answer in standard form. 6 + i 4 − 2 i \frac{6+i}{4-2i}

Now that we know how to add, subtract, and multiply complex numbers, let's learn how to divide them. Well, whenever I divide by a complex number, I'm always going to end up with a fraction that has an I in the bottom. So I'll have something, doesn't really matter what, in the numerator divided by a complex number that has a term with I in it. Now looking at this fraction, I'm not really sure how to simplify it and come to a solution that actually makes any sense. So this I in the denominator is actually bad. I don't know how to deal with it, and I wanna get rid of it in any way that I can. So how am I going to take this complex denominator and turn it into something real? Well, we just learned that if we have a complex number and multiply I'm going to go ahead and walk you through how to use the complex conjugate to divide complex numbers and come to a solution. Let's go ahead and jump right into an example. So I want to find the quotient of these numbers, 3 over 1+2i. Now the first thing I want to do is get rid of that I in the denominator. So my very first step is actually gonna be to multiply both the top and the bottom by my complex conjugate of the bottom. Now since the bottom is the number that I wanna get rid of, I wanna get rid of the I in that, I'm gonna go ahead and multiply by the complex conjugate. Now the complex conjugate of 1+2i, I'm just gonna take that imaginary part and flip the sign. This is gonna be 1 minuteus 2 I. Now, if I'm multiplying the bottom of my fraction by that, I need to also multiply the top of my fraction as well so that I'm not really changing the value of anything. So let's go ahead and expand this out. So with my numerator, I can take this 3 and distribute it into that complex conjugate, which will give me 3 minus 6 I. Now with the denominator, since I have 2 complex numbers, I can go ahead and use foil here. So my first term is that 1 times 1, which will give me 1. My outside term, 1 times negative 2 I, will give me a negative 2i. Then my inside term, 2i times 1, gives me positive 2i. And my last term, 2i times negative 2i will give me negative 4i squared. Now we know that whenever we're left with an I squared term, we can further simplify that. So this negative 4 I squared is really just negative 4 times negative 1, which we know is just positive 4. So let's go ahead and rewrite our fraction numerator is still just 3 minus 6 I. And then my denominator here, let's look at what terms we have left. Well, I have this one, and then I have minus 2 I plus 2 I. But I know that those middle terms are just going to cancel out, so I don't have to worry about them. And then I have this plus 4. So I'm simply just left with 1 plus 4, and 1+4 is just 5. So my denominator here is 5. Okay. So we have completed step number 1. We have multiplied the top and bottom by our complex conjugate and simplified as much as we can. Now let's move on to step number 2, which is actually to expand our fraction further into the real and imaginary parts. So I wanna take the real part and I want to split it from my imaginary part. So looking at my fraction over here, I can just take the numerator and split it, keeping that denominator on both terms. So I will really just end up with 3 fifths minus 6 over 5 I. So I have expanded my fraction into my real and imaginary parts. Step 2 is done. Finally, step 3, which is to simplify our fraction to our lowest terms. So I have 3 fifths minus 6 fifths I, and this is actually already in its lowest term so that means that this is just my solution. And I've completed step 3, I'm completely done, and I have my solution. That's all there is to dividing complex numbers. Let me know if you have any questions.