Hello there. We're going to work with a quadratic function today. We're going to find its complex zeros and we're going to grab it and label it's intercepts. So what does the quadratic function look like? Any quadratic function can be represented using an equation of the form F A bex equals a X squared plus B X plus C. And the graph looks like a parable a U. Shaped graph. But the one that we're going to be the function where we're going to work with um see apart a. We're gonna start by finding it zeroes and it is alfa backs equals X squared minus two X plus five. Okay, so let's start by finding the zeros of this quadratic. Now finding the zeros of any type of function can be done by setting Effa Becks equal to zero. So if I'm setting F of X equal to zero since f of X is equal to this quadratic expression here, X squared minus two X plus five. Set that equal to zero. And now I can solve this equation. I uh you know, we can sometimes factor quadratic expressions into the product of binomial. I don't think these factors. And since we're it's asking to find complex zeros, which could be just real numbers, either rational or irrational numbers, but I have a feeling there might be the imaginary number. And so factoring this is much more difficult. I think it's going to be simpler if we rely on the quadratic formula which is X equals the opposite of the plus or minus square root of B squared minus four A. C divided by two times A. Because any we're going to use that because any type of quadratic can be sold using this formula. So our value of A is one. B is negative to see is five. So let's use that formula to find the zeros. So the zero the X X. Values which we'll find here in a moment. So opposite of the or negative B. B. Is negative two. Then switches to positive two. In the formula, let's remind us nominees Princes Any for any of the other substitution of A. B. And C. B squared to negative two squared minus four times A is one times. See it's five divide that by two times is one. What I do at this point is I take all of this stuff that is in the square root under the square root here, the discriminate plug into the calculator. And so when I do that, you will end up with negative 16. Right or minus 20? Negative 16. Okay, So it was a copy of this expression down except the stuff under the square root is negative 16. Now, what is negative? The square root of a negative that that's allowed. We can handle that. Um We can take square, it's a negative because we have the imaginary number squared of 16. Kind of walk to the side here, square to 16 is four. This word of negative 16 can be simplified as and give it as negative one time 16 square to 16 is four, but the squared of negative one is I have a kind of reverse the order there, but traditionally written with the eye after the coefficient. Um So we have four I is what spread of negative 16 is equal to. So let me go back to this. I have two plus or minus. That's word of negative 16. We just saw as equivalent to four. I all this is divided by two and then I'm going to think of it this way as dividing two by two and four I by two. So where do we end up here? We're solving for X. We're using the quadratic formula to solve for X. Find the zeros and we have two divided by two is one. Let's remind its four I divided by two is two I. So there are these zeros. Um They are complex, they haven't met real and imaginary component. It's actually to this expression represents two zeros, one plus two I and also one minus two. I. So let's go ahead and move on to graphing this quadratic and we'll label it's intercept once we have the graph here's part B. We have same function X squared minus two X plus five does most. Dot com slash calculator is a really a great website to help you visualize the graphs of functions and and these X values that I chose are they're kind of you know in order negative +123 going up by one. But I didn't just randomly choose these. I chose once. That's going to help me help us visualize some night the symmetry of my parabola. Um And I didn't I wouldn't just say why not? Let's just start at negative 10 negative 505 and 10 because if you're trying to see what the graph looks like with a parabola, you can't just randomly choose. Um So using the graphic calculator technology is helpful and then evaluating for X we couldn't take our X. I use and then one at a time plug the X. I was into the expression and evaluate what I used was days most actually create a table and it will quickly give us the Y values. But when X is negative one then we have a Y value of eight. When x is equal to zero. That was easy to see because if you plugged in zero the X. All the X terms cancel out your left with five. You plug in one. You will get four two gives you 53 gives you eight. Again. See some some symmetry here on the table. 85458 Okay. So let's go ahead and grab this. Something will go with red. Have several xy pairs. I'm going up to eight. So let's see. I'm gonna go by twos. Why not have I think on the X axis I can just go buy ones. We're going up to three one two three opposite direction negative one 23.1 negative one. And then on the Y axis. Why don't I can I can scale the axis. However like I think if I went by this let's see. Two 46 Yes, I do want to draw to scale. Okay. All right, so let's go ahead and graph this and we have negative 18 So again, these are 2468 So negative 18 is one of our 10.5 to 4. Hi. Mhm. 14 38 I skipped over 25 And I want to do that to five. That's the thing about making your own graph. It doesn't always end up looking very nice. 38 So we have a parabola. We have an idea we have a one representation of what this graph looks like. And I would recommend using screw up a little bit here using dez most dot com slash calculator. To get a better I think would be a better representation of this graph. But we can see um that it does not cross the X axis and that's because the zeros that we found one plus or minus two. I. Are complex. They're imagine they have an imaginary component. And since our X axis has only real numbers and that makes sense, is not going to cross. So we have to label the intercepts. There are no X intercepts, but there is a Y intercept, which is five gone. Going to go and label that there. So where we have the graph label, it's intercepts. And we had one cluster minus two. I being the complex zeros. Thank you for watching.