 Let's do factoring of quadratic functions, okay? So let me give you a little intro and by the way if there's math questions Let us know make a little list here that we're going to go over but right now we're going to talk about factoring Venetra We're going to talk about factoring quadratics Factoring quadratics Let me give you a little intro Factoring quadratics this thing right here, right? And quadratics are parabolas when they say quadratics in general This is what we're talking about, right? Something that does this right so if you throw a pen You'll do this Right, so this stuff comes in handy in economics comes in handy in kinematics comes in handy multiple different places, right? I always say that Japanese is in many many ways easier to learn than French They're tried okay so Factoring the reason if you look at them have been popping onto the math streams or if you talk to people trying to learn mathematics and whatnot Usually one of the places that people have a hard time with there's a few different places that takes people out of the game, right? One of the places that takes people out of the game is factoring quadratics because this is really the first time You're introduced to the concept of factoring Factoring so it's not factoring quadratics that takes people out of the game. It's factoring That takes people out of the game The reason that most people are taken out of the game when they encounter Encounter factor in quadratics is because they don't understand what factoring really is because it's never really explained properly Or let me paraphrase that right or Say in a way that's not condescending I Found with students that I work with once they understand what factoring is Then they don't have a hard time factoring quadratics because quadratics are just fabulous, right? So what is factoring? Factoring is us Looking at a system and breaking it down to his core properties, right? Factoring can be thought of as the same process as factoring an Integer, okay, or a natural number, right? And if someone's doing factoring quadratics, they've already encountered prime numbers Factoring natural numbers, right? So consider this consider the number 28 28 now one of the things that Happens is when you're coming into high school or what not or looking at the real number set What you find out is this is called a composite number It's not a prime number and prime numbers and numbers that can only be divided evenly by one and themselves, right? So 28 is made up of other numbers, okay? So what you can do is break down 28. So take 28 and break it down into Things that multiply together to give you 28 If the number is even you can always divide it by two. So you're gonna take 28 divide it by two two times 14 gives you 28 Right, and this is multiplication between these but we never put it in because we know it's multiplication We're talking about multiplication Now two is a prime number. It can only be broken down into one and itself Multiply it together to give us two so anything that Fits that pattern that can only be broken down into one times itself to give itself Then we don't break it down, but 14 is two times seven, right? So we break this down two times seven cool, right So what we find out is 28 28 is really two times two times seven cool useful Damn right, it's useful. It's very very useful. It's on the same concept of us Taking something in natural world Right and breaking it down to find out what its building blocks are Right, for example take water Okay Water So take water Mask of a raven. Let's check it out when I first explained to one of my friends the actual purpose of going from For example that to that rather than just a trick. He'd learn he found it very illuminating very illuminating indeed Indeed, right so on the same concept of This number right take water. What's water water is each to oh, right each to Now back in the day as human beings Before we understood what? molecules atoms During the time where we thought everything in the world was made up of five elements earth wind fire water and Wood or soil, I don't know what it was right, but one of them has always been water during that period We looked at water and what that's water. We thought water was just water Nothing made up water water was the thing right and then as we evolved as we understood more about the world We realized that water is really pure water is really each to oh cool What is each to oh break down each to oh each to oh says There is each to two hydrogens and an oxygen Cool, right and two hodge hydrogens means hydrogen and a hydrogen Cool, what water is two hydrogens and an oxygen? Cool. Why is this important? Well? What else can you make with this building block right because if you're building block your core building block is each to oh You have to make whatever you make has to have each to oh in it, but if you're able to break this down into his prime factors right prime Elements then you can take parts of these and create something else right you could take a Two from 28 and a seven from 28 Two times seven is 14 multiply it by five. What do you come up with cool? Zero two you come up with 70. Oh, that means if we break down 70 It's seven times ten Ten is two times five So it's two times seven times five two times seven times cool We just came up with a new number right guess what we do the same thing with these things, right? Guess what we do the same thing with functions right So on this same concept We can take any function and try to break it down into score building blocks, right? Let's start off with one of the easy functions. Let's start off with a quadratic So it's breaking down the thing called factoring. That's exactly what it's called void, right? Breaking down the thing is called factoring. It's called breaking it down to see what its core building blocks are That's it and a quadratic is just a function. That's a parabola that does this Guess what we have an infinite Number infinite types of functions. Here's a quadratic A cubic does this power four does this Power five can do that You can have a line this guy is really one of the core functions. We have right You could have a function that does this, right? There are an infinite number Types of functions we have and what we can do is we can factor all of them Or we can try to factor all of them and a line is Usually considered to be the prime Factor that prime function of a lot of these Functions, okay So the way we get into factoring for us the first thing we factor Try to factor is a quadratic because a quadratic is a parabola and a parabola is really two lines multiplied together, right? So in my part of the world, they teach you the lines in grade 9 if you're lucky not really they don't teach you that if you're lucky you got a good teacher They teach you that in grade 9, but usually in grade 10 They encounter this in my part of world in other parts of the world. You probably do this in grade 5, right? Some parts of the world you don't even get a chance to do this, right? Straight line is called a linear function, right? Exactly. Why is this called a linear function? Or the way you can remember is called a linear function. I don't know why it is, but it's called a linear function because It's a line, right? So consider linear functions to be your prime Numbers if you were breaking down Natural numbers, right? So first thing we're going to do is we're going to look at a quadratic, right? So let's take a look at a quadratic Let's take an example of a quadratic function Now if you've been here, you know, there's one quadratic function I like, right? Over the real number There are polynomial factors of any degree. For example X squared, X plus 1 cannot be factored and acts as a sort of a polynomial prime. Exactly. Yeah Right? Get into that later down the road Right? It doesn't necessarily have to be a line Sometimes you can't break things down any further, right? And those become your prime functions, right? It's not as it's not as simple as prime numbers Prime functions, right? So take a look at this. Let's take a quadratic. Let's take this quadratic I'm going to use f of x. If this is freaking people out Think of this as y because that's what all it is, right? So if you want, we can just write it as y Y is equal to X squared plus 5x plus 6 Okay So let's assume you're given this polynomial function, right? The polynomial is just a smooth function Quadratics is one of the core base Polynomial functions, right? So let's assume you're given this quadratic function And it's a quadratic because this thing graphs a parabola. We talked about. We can talk about it further, right? Try to graph it and stuff and we've done a lot of this quadratic Graphing quadratic functions. If you look up Ciccio completing the square, you'll see us Graphing these guys, but right now let's assume We were given this function and this function explains some system in our world, right? And We wanted to break it down. We want to find out if there are any prime Functions within this function Right, what's this thing made from, right? And this is Called a simple trinomial, right? So what you do with simple trinomials because there's no There's only a one in front of the x-square. You look for two numbers that multiply to give you six and have to give you five Yeah, in the complex numbers every polynomial is fully factor factorable Factorizable into linear factors. Awesome. I got to look further into this one, right? So if we're trying to factor this quadratic simple quadratic function A simple trinomial We look for two numbers that multiply to give you six multiply to give you six And add to give you five Add to give you five Okay, and always remember the sign in front of the number goes with the number, right? So this is positive six and this is positive five Okay Always start off with the multiplication part because there's less Integers, there's a finite number of integers that can multiply to give you This number then add to give you this number okay, or Yeah integers, so there's a finite number of numbers Integers that can multiply to give you that, but there's an infinite number of of integers that can multiply to give you the middle number or add to give you the middle number. So don't start off with the middle number, start off with this number. List all the numbers that multiply to give you six. We got one times six, negative one times negative six, two times three, negative two times negative three, right? One times six, six, negative one times negative six, six, two times three, six, negative two times negative three, six, right? So what you do now is, hopefully this is coming out okay for you guys. Now what you do is add these two numbers. One plus six is seven, so that doesn't work. Negative one plus negative six is negative seven, so that doesn't work. Two plus three is five, that works. Let's check to make sure the next number is not going to work, right? Or what it's going to equal. Negative two plus negative three is negative five. We're looking for positive five, so that's not going to work. So the two numbers that multiply to give you six add to give you five or two and three, right? Yeah, two and three or three and two. So all you do for this is you go, oh, this guy, you can break down into xx plus two plus three, right? So this broken down into prime factors is this guy times this guy. That's cool, right? These are lines, linear functions, each one of these, right? So if we call y1 is equal to x squared plus five x plus six, let's give this guy a name. Let's call this w. Let's give this guy a name. Let's call this z. So y is really w times z. Now you don't have to go there. I'm just doing this for visualization. And what's w? w is x plus two, z is x plus three, right? We're just using substitution basically. God, I miss being taught something fascinating. So why? This is what we found out so far. Why can break down into w times z? You can think of these as different elements, right? I miss mathematics. I'm so glad math is part of my life, really. As someone who has been, his life has been revolved around mathematics for the last couple of decades at least, right? Man, I couldn't imagine my life without mathematics. It's good for the processing system, really. It is an amazing BS detector. It's fantastic to meditate to. It's fun to play with. It improves your abilities to do whatever you want to do in life. It gives you better understanding of the world. It's win, win, win, win, win, win, win, win, right? Tangpo, how are you doing? Yeah, it's amazing, right? So consider why. We took y. And for us right now, y was dysfunction. But it could be H2O. It could be water. It could be a number. It could be another compound we're breaking down. We broke y down into a w and a z. Wow. Y is made up of w and z. W, what's w? w is x plus 2. And z is x plus 3, right? I'm going to do a little cleaning house. And what we're going to do is we're going to graph this, okay? Howdy chat. Amigos. Right? So let's break this down. I'm going to erase these guys. We're going to need more space, right? So let's break this down or erase this. And write down the core stuff that we know we found out about this. We found out that this thing is x plus 2 and x plus 3. Is that big enough for you? Oh, yeah, it's big enough for you guys to see. And what did we say? We said x plus 2. We called w is equal to x plus 2. And we said let z equal x plus 3. That's a linear function. That's a linear function. A line and a line, so let's graph both of these lines. Now, I'm going to assume you know how to graph a line. Voyager 2. How are you doing? This is the x-axis. That's the y-axis. Now, you know what? Let's graph it using a table of values for this one. And then we're going to straight up graph it there if you know y equals x plus b. Let's use a table of values for this. Now this guy, this axis, right now we're going to graph w. And then we're going to graph z. And then we're going to graph y. So if we're using a table of values to graph this, x and w. Just plug in numbers for x and find out what w is. So the first number, usually easy number to deal with, you can plug in 0 for x. If you plug in 0 for x, this becomes 0 plus 2, which is 2. If you plug in 1 for x, you're going to find out this is 3. Let's grab 1. We've got 1 here. We've got 1 here. Let's grab 1 on this side. Negative 1. Negative 1 plus 2 is 1. And if you have negative 2, negative 2 plus 2 is 0. Let's plot these points on the graph. Right? 0 and 2. 1 and 3. 1, 2, 3. 1 and 3. Negative 1 and 1. Negative 1 and 1. And negative 2 and 0. Negative 2 and 0. So this line, w, looks like this. This is the line w. Let's graph this guy. Okay. I'm going to erase our table of values here. If we're going to graph, here I'm going to circle this so we know what we're graphing. That's w. Here's z, right? So let's graph z. If we're graphing this, we can use the function form notation, y is equal to mx plus b. Where b is your y intercept, m is your slope. So if you know how to read this, and it's a sentence, by the way, mathematics, all of the stuff that we write down, they're sentences. They're saying something. They're telling us something. So z is equal to x plus 3. That's the y intercept. And the slope is 1 over 1. So here's the y intercept. And you go up 1 over 1, which is the same slope as this guy. So you can make a line that's, whoops, let me make up. Here is z. The line z. So we have two lines. This guy's w and that guy's z. Okay, cool. Now, this is what this factor tells us. If you notice this, we have this guy times this guy, which is w times z, right? Which means if we take w, this line, and multiply it by this line, we get that guy. And the graph of that guy, let's graph it. Should I use a different color? Let's use a different color. Cool. Let's use this color. So we're going to graph this. Okay. The graph of this guy, and this is negative 3 here for that guy, is going to go through here. Here, let's graph it using a better color. Let's graph it using brown. How did you get the slope? The slope here, here, let me do a little aside, fill in the blanks. So equation of a line is y is equal to mx plus b. b is your y-intercept, and m is your slope. All right? And slope is rise over run. Right? Rise over run. Okay. For this function, we had z is equal to x plus 3. If there's no number in front of a variable, it's just 1. Right? So this is 1. And we could always make a fraction out of any number by putting it over 1. So 1 over 1. So the y-intercept for this is y int is equal to 3, which is here. And the slope, slope, which is m, is 1 over 1. So we went up 1 over 1. Does that make sense? I hope so. Let's graph this using brown. Logical. I love it. Nice. Cool. Yes, good description. Okay. Now take a look at this. We're going to graph this guy. Now, should we kick it out one more, let's kick it out one more level of complexity so we see really what's going on. Right? I'm going to erase these guys. Well, let me paraphrase. We're not going to kick it out one level of complexity. We're going to give it a little bit more explanation as to why certain things are the way they are. Right? Now I have a video out there. Let me get you this video. Okay. And it's called the power of zero. Cheechoo. Power, oops. Power of zero. We've done a couple of these, but here, I'll give you this one. Take a look at this video. Full reference anyway. Now in this video, I'm about to let you know what it is that we're doing in this video. What the meat of that video is. It's going to take us a minute to explain it. Now zero in mathematics gives us problems. We can't divide by zero. If we divide by zero, the universe explodes. But zero also provides us solutions. Here is a question for you. Let's say we have a times b times c times d equaling zero. How can you multiply four things to equal zero? What can you deduce? What can you conclude about a, b, c and or d? Right? At least one needs to be zero. Right? At least one of them needs to be zero. Very important. Or all could be zero. Yeah, all could be zero as well, for sure. Right? But at least one of them has to be zero. That's not the case if this was two. Right? Or any other number than zero. If this was two, you couldn't say at least one of them has to be two. That's not correct. The combination of a, b, c and d multiplying together to give you two is infinite. Right? The possibilities are infinite. Okay. However, as soon as we set this equal to zero, then at least one of them has to be zero. Wow. We just took it infinite, something that was infinite and reduce it down to at least one of them has to be zero. Powerful, powerful, powerful. Right? Incredible. Cool. Well, how does this apply? Here. We'll take a look at this. Right now we have y is equal to x plus two times x plus three. Right? Huh. Now let me ask you this. This is a function that plots on an x y axis. Right? On an x y grid. Right? So ask yourself this. When does this function cross the x axis? When is this function going to cross the x axis? Just the same way you could ask this. When does this function cross the x axis? And when does this function cross the x axis? Right? Well, this function, w crosses the x axis when w is zero because w is this. This line is both are or it's y, w and z. Not when x is negative but when x is zero. Yeah, when x is negative for these ones, my apologies. Right? But not just any negative. It has to be a certain negative, negative two. So if you want to find out when this crosses the x axis, you just set w equal to zero because this is w equals one, two, three, negative one, negative two. So this has to be zero. Right? That's our scale. So all you do is just set w equal to zero. This becomes x plus two. Bring the two over so x equals negative two. That's what it is. That's where we are. Negative two. Let me use the right colors for this. So this is negative two. And this guy here, if you do it for this one, set z equal to zero, bring three over. That's negative three. Right? Cool. What's my math background? I got my degree in geophysics and a minor in mathematics and I've been teaching math for like 20 years plus high school mathematics only. Right? It's not very high. I just know how to teach high school math. Now, what about dysfunction? Well, we haven't graphed dysfunction yet. Right? But we can ask ourselves, okay, just curious. No worries, Bacon. Bacon slaying. Welcome to our live stream, by the way. So we have dysfunction. y is equal to x plus two times x plus three. So we haven't graphed dysfunction yet. But let's ask ourselves, when does this function cross the x-axis? Well, this function crosses the x-axis when y is equal to zero. So let's set y is equal to zero. Let's take this guy, x is equal to, or sorry, y is equal to x plus two times x plus three. So we're going to set y is equal to zero. Right? When y is zero, we're on the x-axis. So we're asking ourselves, when does this function cross the x-axis? Well, we take this and link it up with here. We have two things multiplied together to give us zero. How is that possible? The only way that's possible is if one of them is equal to zero. Okay, so you set each one equal to zero. So you can say, oh, this is true only. This function crosses the x-axis only when either this x plus two is equal to zero or x plus three is equal to zero. That means x is equal to negative two and x is equal to negative three. Oh, wow, cool. It's at the same place. Wow, nice. So this function, this function crosses the graph here and here. We don't know how it looks aside from that, but we know that it crosses the x-axis here and here. Okay, very cool. Very cool. Now, we can graph this using Completed Square, but I don't want to do that. I want to show you another way you can look at this whole thing. This will blow you away a little bit too, right? Or it should solidify your understanding of this concept. Take a look at this. Let's create a table. Right. And by the way, this graph, let me give you a function. Do you have permutations and combinations in high school? We do. I'm not very good at it with permutations and combinations because they play word games with people, right? Factoring. Let me see if I can find it with one. There it is. Here's a video. It's called Factoring Polynomials, A Graphical Representation, Why We Factor. And I put this video out. This is part of the Language of Mathematics series, and I put this video out in 2010, almost 10 years ago. Cool. Right? Nine and a half years ago. And this is the function that we talked about. It's an eight minute video that'll graph it for you as well. Right? But I'm going to show you a table format of how you can take a look at this. Appreciate what's going on here. Right? Now take a look at this. Like, wow, 2010. And that was me at a skate park with my tripod and people skateboarding around me and me doing mathematics on the walls. Right? You were ahead of time. I just did what I did. Right? Now take a look at this. Let's create a table. X, X, Y. Actually X, W, Z, and Y. Right? X, W, Z, and Y. Okay. This is W, that's Z, that's Y, and our X appears in all of them. Right? It's the independent variable that both or all three of them. W, Z, and Y are dependent on. Right? So X's are independent variable and W, Z, and Y are dependent on X. Right? So let's find out what W, Z, and Y are for certain values of X. Cool? Cool. Let's plug in X is equal to zero. What's W when X is equal to zero? You put zero in for X. So W is equal to two. Z, if you put in zero for X, Z is equal to, because this disappears, Z is equal to three. If you put in zero for X here, this becomes two. That becomes three, two times three, six. Right? Do you see what's going on? Let's do another number. Let's plug in X is equal to two. Okay? Should we do two? Yeah, let's do two. Let's do one first, so we don't go too far. Oh my god, at least one sense. One sense. Wow, nice. So take a look at this. Let's put in X is equal to one. One. Plug in one here. One plus two is three. So W becomes three. Put in one here. One plus three is four. Right? What's Y? Put in one here. One plus two is three. One plus three is four. Three times four is 12. Wait a second. Wait a second. Exactly. And Y is just these two multiplied together. What? Right? Let's put in X is equal to negative. Negative. Let's make it not easy for us to multiply. I don't know what that says. Let's put in negative eight. Negative eight. When X is negative eight. Negative eight plus two is negative six. Negative eight plus three is negative five. Negative six. Well, if you put in, you could put in negative eight here if you want, but we don't have to anymore. We could just multiply these two. Negative six times negative five is 30. That's what Y is. Wow. 30. I think negative six times negative five is 30. Right? So check this out. When X is zero, W is two. Z is three. Y is six. One, two, three, four, five, six. When Z is one. Oh, sorry. When X is one, are you steady? We're doing. We're just doing math, learning math, practicing math, meditating. When X is one, W is three. When X is one, W is three. Well, it was. We knew that. Z is four. Well, we knew that. Y is 12. Oh, man. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12. I've been out of contact with any school subjects for nine years now. No, you're not, man. You're not. It'll take you two seconds to get this back. The cool thing is, I don't want to. Hi, Gicho. Hello, QC warrior. How are you doing? How is life? Right? Negative eight. Negative eight. We're not even on the board. Negative four, negative five, negative six, negative seven, negative eight. When X is negative eight, W is negative six, right down here. Z is negative five, like down here, if you extend this. And Y is 30, right? What does the graph of this look like? This parabola look like? This train is about mathematics. It's easy to review math unlike some other subjects. Yeah, math should be the easiest course you take in high school, period. University, different game. High school should be the easiest course you take because it just builds from previous years, right? 30 is definitely out of the web board. I think some better, right? If we end up graphing this function, this is what it's going to look like. Take a look. That's what it looks like. Let me make this darker so it comes out. My pens are running out. I need to go to the stationery store. We need to go get ourselves more stationary. Having a little break from steady pun intended and I saw you were live. So thought I would take my little break here. Awesome QC warrior. I'm glad. It's a nice place to show. So that's the function. Now, here's the kicker. To me, this is the kicker, right? People don't realize, but this graph stuff is really important in engineering stuff. Huge humongous. It's crazy important in economics as well. Crazy important everywhere. Crazy important everywhere, right? And here's the kicker. We took two lines, right? Then we two lines multiplied them together and we got a curve. Why is that? We got a curve because of the property of mathematics where if you multiply two negatives, you get a positive, right? Even in economics, right? So if you multiply two negatives, you get a positive. Now take a look at this. Now I talk about this in the short video, the eight minute video that I linked up previously, right? When we're breaking the sucker down, okay? But think of it this way. The zero point, the factoring. When you're factoring, you're finding where functions cross the x-axis. That's what it is. Long story, right? Why do we factor? We're factoring to find where functions cross the x-axis, right? In the most simplistic form. Sometimes you're factoring to find out when functions cross each other, right? And these linear lines have endless possibilities, right? And they go on forever, yeah. Unless we give it boundaries, limits, right? That makes sense according to our systems, okay? But take a look at this. Let's assume, oh, this is going to be red. This might not come out well. I'm going to try green, this strong green. So the x-axis is sort of an important point, right? So take a look at this point. Let's extend this up. And let's extend this up. This is basically the line x is equal to negative two, and this is what x is equal to negative three. If you look at these functions, all three functions, w on this side of negative two, x is equal to negative two, is positive because it's above the x-axis. What z is positive because it's above the x-axis, right? So positive times a positive gives us a positive. So we're above the x-axis, right? If you look at this zone from negative two to negative three, w is below the x-axis, so it's negative. z is above the x-axis. In this zone, z is above the x-axis, so it's positive. Negative times a positive is negative. That's why negative times a positive is negative. That's why our function y is below the x-axis, right? And then as you get closer to negative three, this number gets smaller, so it's kicking up this y-value. And then once you go to this side of negative three, when x is negative three, both the z and w are below the x-axis, so they're both negative. And negative times a negative is positive. Positive. So when you're multiplying two linear functions, when you go below the x-axis, there are two negatives multiplying each other to become a positive if you're talking about quadratic. So our function y has to start coming back up again. It has no choice. That's its behavior, right?