Quadratics & the Fundamental Theorem of Algebra

let's say that we have the function f of X being defined by the second degree polynomial 5x squared plus 6x plus 5 the fundamental theorem of algebra tells us that because this is a second degree polynomial we are going to have exactly two roots or another way of thinking about it there's exactly two values for X that will make f of X equal zero so I encourage you to pause this video and try to figure out what those two values of X are so when I if we're looking for the x values that would make this expression equal to zero that's essentially trying to solve this equation at 5x squared plus 6x plus five is equal to zero and there's no obvious way I can think of factoring this right over here so I'll just resort to the quadratic formula so the quadratic formula tells us that X where X is a solution to this equation is going to be equal to negative B I'll try to color code it negative B this is right this is B right over here negative B so negative 6 plus or minus plus or minus the square root of B squared of B squared minus 4 times a times a times C times a times C all of that all of that over all of that over 2 times a all of that over 2 times a so what does this simplify to this is going to be equal to so let me go back to that this is going to be equal to negative 6 I'll try to keep it color-coded still negative 6 plus or minus plus or minus the square root now what is this going to be equal to this is 36 minus a hundred so negative 64 all of that over 2 times 5 all of that over 10 now this is interesting we are trying to take the square root of a negative number or another way of thinking about it b squared minus 4ac is less than 0 b squared minus 4ac which is often time oftentimes called the discriminant this of this quadratic this is less than 0 so this is less than 0 this part of the quadratic formula we're going to try to take the square root of a negative number this is going to be a negative number so we're going to result in an imaginary number and so what that gives us they are not two real roots but two non-real complex roots so this is going to be equal to negative 6 plus or minus 8i 8i the square root of negative 64 is 8 I if we extend the the principal square root function to imaginary numbers or complex numbers all of that over 10 or we could say X is equal to let's see if we do if we find a greatest common divisor if we try to reduce this so we're gonna have let's see they're all divisible by 2 so it's negative 3 over 5 that's the same thing as negative 6 over 10 plus or minus 4 over 5 4 over 5 I or you could say that the two roots which are non-real complex roots so it's X is equal to negative 3/5 plus 4/5 I that's one of the roots and then the other root is X is equal to negative 3/5 minus 4/5 I notice we satisfied the fundamental theorem of algebra we have two roots they're non real but that fundamental theorem of algebra says hey look we're just gonna have at least if we're nth degree polynomial we're gonna have n complex roots they could be real or they could be non real and we see that right over there we also see that they are conjugates and the quadratic formula kind of sets up a situation where we are especially if this ends up being less than 0 and so this when you take the square root you get an imaginary number you see where those conjugates appear now let's verify graphically that this is indeed the case that this does not have any real roots so let's get a calculator out so let me go into graph mode so Y is equal to let me clear what I must have been doing out here before so y1 is equal to 5 times x squared plus 6x plus 5 and then let me set a reasonable range here so I don't know actually I know very little about to this function right over here so I'm gonna set a minimum range and an egg ative 10 x max I don't know positive 10 X scale as well let's see Y max let's see this thing actually does get pretty big pretty fast so let's say Y max is the y scale I don't know I'll say a hundred a hundred the scale I'll make it 10 each of those notches are going to be 10 Y minimum well it's we want to see the x-axis to make sure it doesn't cross it so let's go at negative I don't know negative 20 and now let's graph this I'm hoping that I've actually captured it and I have so there you have it you see that this thing does not intersect the x-axis and in fact we could we could zoom in on it let's see it's a little bit hard to it's a little actually let me just change the range a little bit so let me just change the range so let's make our X min let's make it 5 let's make our x max whoops not 50 let's make that 5 and let's make negative 5 and positive 5 and then let's see let's make our Y max let's make our Y max equal to 20 and then our Y scale I know we can make it to okay I think this will get us much closer in to there we see it does not intersect the x axis this does not have any real roots but it has two non-real complex roots