Evaluate (x^2 + 19)/(2 - x) for x = 3i.

Question

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we're going to be plugging in X. Is equal to two. I into the expression X squared plus 12 divided by three minus X. So if I do that I have to I squared plus 12 divided by three minus two. I notice how by plugging in to I for X. I am making this expression deal with complex numbers where we have real parts Such as the three And the 12 and then we also have imaginary parts wherever I plugged in X. So if I do this multiplication in the numerator where I have to I times two. I I'm left with four I squared plus divided by three minus two. I but what does it mean if we have an I squared recall that I is equal to the square root of negative one. So if I have I squared that is the same as the square root of negative one times the square root of negative one. And whenever we have two square roots being multiplied you're left with just what's on the inside. So that's the same as negative one. Which means that wherever I have I squared I can substitute negative one. So you can see that by squaring the I I no longer have an imaginary component. I'm left with the real number of negative one. So if I do that my numerator seems to be the only one that's affected where I have four times negative one because I'm replacing the I squared and everything else stays the same. So this becomes negative four plus 12, divided by three minus two. I Or eight divided by 3 -2. I So now I'm left with eight divided by 3 -2. I However whenever we're doing division with complex numbers recall that we can't have imaginary components in the denominator and in this case we have A. Two I. Right here that makes the denominator have an imaginary component. And the easiest way to get rid of imaginary components in the denominator is to multiply by the complex conjugate. So recall that if I have a number such as A plus B. I. Then the complex conjugate is simply found by flipping the sign between the real and the imaginary part. So if my number is A plus B. I then the complex conjugate would be A minus B. I. So going back to my denominator, the complex conjugate for 3 -2, I would be three plus two. I because I'm flipping the sign between the real and imaginary parts. However, you can't just multiply the denominator by three plus two. I I also need to multiply the numerator because three plus two I divided by three plus two. I is the same thing as multiplying by one. So we're not introducing any new variables into the expression. And if I do this multiplication, I have eight times three 24. eight times 2 i divided by three times times positive two. I I got six I negative two I times three negative six I negative six I times positive two I is negative for I squared. And you can see that these middle terms in the denominator cancel out because positive 6 -6 is zero So I don't have to deal with those anymore. And I'm left with 24 plus 16. I divided by nine minus four I squared. We'll notice that we still have this imaginary component in the denominator but recall that we solved earlier that I squared is the same thing as negative one. So let's go ahead and plug in negative one for I squared. And if I do that I'm left with 24-plus 16 I divided by nine plus four. So you can see that my denominator no longer has any imaginary components because nine plus four is 13 which is a real number. So I'm left with 24 plus 69 divided by 13. Now I need to separate my fraction into the real parts such as the 24 and the imaginary parts. And if I do that I'm left with 24 divided by Plus 16 I divided by 13. So this becomes my final answer. And you can see that this aligns with answer choice A So hopefully this video helped and see you next time

Evaluate (x^2 + 19)/(2 - x) for x = 3i.

Key Concepts

Complex Numbers

Polynomial Evaluation

Rational Functions

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