Evaluate x^2 - 2x + 2 for x = 1 + i.

Question

Accepted Answer

Hello everyone in this video we're going to be evaluating the following polynomial X squared minus four X plus five where X is equal to two plus I. So in this case you can see that we're dealing with a complex number because X has both an imaginary component and a real component. So I'm gonna go ahead and plug in this value of X into my polynomial. So my equation becomes too plus I squared minus four times two plus I plus five. And if I write out that squared person I have two plus I times two plus I getting rid of the squared because I'm writing everything out minus four times two plus I actually let's go ahead and just distribute this four. Since we're already rewriting this I have negative four times two is negative eight, negative four times I is negative four I and then plus five. So you can see I now have the complex numbers being multiplied in the front and then I have some terms back here that I can combine. So if I do this multiplication, I have two times two is four. two times I is too I I times two is 2. I I times I is I squared and then I can combine the negative eight with the positive five and get negative three minus four. I So now I'm going to try to combine My real numbers together and my imaginary numbers together. So in this case I have negative three and 4 I can combine those and just get one and then I have to I two I and negative four I and that becomes zero because two I plus two I is four I and then we have minus four over here. So this is zero and then I'm left with an I squared. So this becomes one plus I squared. And whenever you're working with complex numbers it's important to remember that I is equal to the square root of negative one. So if I do I squared I have the square root of negative one times the square root of negative one. And that leaves me with just negative one. So you can see that I started off with an imaginary number but negative one is a real number. So whenever I get squared it turns into a real number. And if I do this, if I substitute negative one into any I squared component, I have. My equation becomes one plus negative one And that leaves me with 1 -1 which is zero. So this is my final answer. And you can see that this aligns with answer choice B. So hopefully this video hope didn't see you next time

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