 So what we got is x squared is equal to 4 and we solve for that by taking the square root of both sides or by factoring So what we got is x is equal to plus or minus 2 now As we talked about before What this is is just one box something equals another thing all it is It's just one function function going on this side, which is four and one function going on that side Which is x squared right so each one of these on either side represents a function Graphically if you want to if you want to visualize that we're going to do it But right now all they are is just functions right this function over here is x squared and that function over there is Just just four and x squared as a parabola it grabs a parabola as a quadratic equation And that one is the function equaling four. It's just a it's just a horizontal line Okay, and we'll talk about the stuff when we get into the polynomials I'll start graphing polynomials and Just doing our terminology all I did when we talked about the you know the terminology of polynomials All I said was just f of x is equal to f of x is equal to x squared Okay, so that's one function and f of x is just our y term right and that one is h of x is equal to Four which is you know a symbol for a different function. I could have called that one f of x I could have called that one h of x I'm just you know using f and h just to make sure we distinguish between the two right now Let's graph that function and then we'll graph that function as well and you're gonna see that all this is saying is Is asking the question when does f of x Equal h of x when does f of x cross h of x when are the x and Y coordinates in this function equal to the x and y coordinates in this function, okay So what we have here is just the x y coordinate system right the Cartesian coordinate system So what we're gonna do we're gonna graph f of x is equal to x squared and f of x is equal to x squared and we haven't gone into graphing quadratic equations yet But one way you could do it is you could do a table of contents where you you know plug in values for x and Just do it and figure out what f of x is and you know some you know initially when they start teaching about functions That's what you end up doing right so what you could do is do So what you could do is plug in values for x in your function So over here for x equals zero you're gonna go f of zero and that's the bad That's the term knowledge that you end up using and we'll talk about this a lot further, okay But all you do is just say F of zero so when you plug in x squared so zero square is going to be zero So f of x is going to be zero when x is zero So basically means y is zero when x is zero when you plug in one for x one squared is going to be one And so that means f of one is going to be one So why is going to be one and if you plug in two two squared is going to be Four so when x is two y is four Now that's also true for when x is negative one so for the x term We're testing five different points. Okay. We're gonna go plus and minus one plus or minus two So when x is positive one y is positive one when x is negative one y is positive one same with two Positive two gives you four negative two gives you four because a negative number squared is just positive, right? It's just four so what we're gonna do right now is graph that Those coordinates on our Cartesian coordinates system, okay? Zero zero becomes is here X is one y is one That point is there when x is negative one y is positive one when x is positive two y is Positive four one two three four When x is negative two y is positive four and what you do now it's just Draw your parabola connect the dots right our first function Which is f of x is equal to x squared and that's just a parabola that goes through the vertex, right? I'll go through What do you call it the cross hairs? I forget what the term is Goes through the origin. So it's just a parabola that goes through the origin, right? Let's graph h of x now now h of x is just a line all that means is Y is Four everywhere, okay? Now some people have a hard time with this The way they do it. I'm gonna do this in green. We're gonna do this in pink, okay? So all that means is h of x is equal to four It means everywhere y is four irrelevant of what x is so if x is zero Because there's no x term on the side, right? So if x is zero y is four x is two y is four x is infinity y is four x is negative infinity y is four Right, so the way you just graph this it's just a horizontal line on the y-axis So why is four one two three four? It's not just a dot because it was just a dot that would mean x equals zero y-sport It's y equals four everywhere so it becomes this So what we have right now we just graph h of x is equal to four, right? And the y-axis is both h of x and f of x, right? And what we end up getting is we're answering the question of What is x squared equal to four? It happens when at those two coordinates when the two lines two functions cross each other, right? And the coordinates for those functions is x is one two and y is four is Find the x points where x squared is equal to four Which is really just a function, right? So this is the most simplistic form But when you get into more complicated functions, what you can do is have Different shapes different polynomials or different, you know, they don't have to be positive. There's different functions Crossing each other and what you're doing is when they give you something extremely complicated on this side I'm extremely complicated on this side, you know when a whole bunch of things over here equal a whole bunch of things over there All they're asking you is what is this function? When does this function cross this function and sometimes you get multiple answers to this sometimes for our case right now We ended up with two solutions, right? Because of parabola crossing a line crossing a parabola it crossed out of two points Now the line could maybe cross out of one point if it was asymptotic to it Sorry tangent to it, right? Or it could cross it or might not cross it if the line was down here For example if you had x squared is equal to negative four Well, if you brought actually let's do that one, right? If you had x squared is equal to negative four if you brought the negative four over it becomes x squared plus four Equals zero and as we talked about before you can't factor two things that are added together You can only factor two things that are subtracted from each other. The reason being is If on this side if you had x squared is equal to negative four if you took square root of both sides Based on the real number set because we're just functioning in the real number world right now The real number realm right now. You can't take the square root of a negative number and what would happen is function over here H of h of x would actually be h of x is equal to negative four So your whore your horizontal line would drop down here That means the line would not cross your parabola so there would be no solutions for that, right? now Let's let's just go lay out a more complicated question All right, and I don't think we'll solve it, but we'll just lay it out just to explain what it You know what it might look like