 So we've been going off on a sort of a factoring binge right now, and we're just talking about the different types of factoring techniques that we have specifically called anomials, right? So we've already talked about how to factor just numbers in general, anything from the rational number set with prime factorization, right? We've talked about the greatest common factor, we've talked about the difference of squares, we've talked about simple trinomial factoring, and we've talked about complex trinomial factor. We've got two more factoring techniques to talk about, one of them is the quadratic formula and the other one is synthetic division. Now these two, you know, kick it up a notch. They give us a lot more power. So the first one we're going to talk about is the quadratic formula, and the quadratic formula, it's literally, it's the formula that I'm going to take to the grave with me. It's the only formula I remembered years after high school. There was a period in my life where I didn't use mathematics that much, and you know, when I started going back into mathematics, it was the only formula that came to me automatically, okay? More so than even the area of a circle or a circumference of a circle or area of a triangle, okay? This formula, a lot of people freak people out. It looks horrendous, but it's really simple. All it is is just you're taking numbers from your quadratic equation and just plugging them in. The coefficient, and what it does, it gives you the factors, right? It gives you the x-intercepts, your roots, right? And the formula is basically, you know, you should say it, it's x is equal to negative b plus or minus a square root of b squared minus 4ac over 2a, okay? It should roll off your tongue. You should dream about this or have nightmares about this, okay? This is the formula, x is equal to negative b plus or minus b squared minus, a square root of b squared minus 4ac over 2a, okay? All those letters you see there, those are your a, b and c, your coefficients and your constant from the quadratic formula, right? Or any, from the quadratic equation, right? Or quadratic expression or quadratic function, right? And your x there represents your x-intercepts, right? Your roots. So the way it works is you have your quadratic equation, right? Your quadratic function, basically. So you basically, I use the function notation, right? So you have a quadratic function and if this is a quadratic equation where you're trying to solve a quadratic equation, all you do, this f of x disappears and all you do is just say equals to zero. And what that is, the equals zero, the zero represents your y-coordinate, basically, where y is zero and that's where you're intersecting the x-axis, right? So a, b and c in the quadratic formula are just your a, b and c from your quadratic function, from your quadratic equation, okay? So whenever you're given a function like this, an equation like this, you put it in this form and we've talked about this, right? You got to put it in the right order, right? Highest x first and then descending x values and this is basically a trinomial, right? So quadratic formula is used for quadratic equations, but it's a little bit more than that because you can use it for equations, functions, that are ax to the power of 4, 6, 8, where this power, the x squared power, has to be double this power, right? So you could have a function in the following form and it really quickly changed the minus c to plus c because the notation is basically plus c. So for your quadratic formula, for your quadratic equation, you can have ax2 plus bxn plus c, where n is the element of the natural numbers, right? So in here where you got bxn, the n can go 1, 2, 3, 4, all the way up to infinity and the x power in front ax2n basically means that has to be double this guy. So what you end up with is the 2n basically means that this power here always, always is going to be positive, right? That's these types of equations you can use the quadratic formula for and we're going to talk about these things, right? And basically what we're going to do, we're going to do examples where we're given equations like this and we're going to solve for them and find your x intercepts and your roots, okay? Now one thing to keep in mind, and we're going to talk about the quadratic formula a lot more when we get into factoring polynomials, but one thing to keep in mind inside the root symbol, this guy here, that guy is called the discriminant, okay? So what we have here is b squared minus 4ac. This thing is referred to as the discriminant, okay? And the discriminant in mathematics is it gives us information regarding the nature of the roots for polynomial function or polynomial equation, okay? Right now we're just going to talk about b squared minus 4ac. There are discriminants for, you know, cubic functions, quartic functions or whatever, right? It's just a general term and it refers to, for the quadratic formula, it refers to b squared minus 4ac. And this thing, whatever is inside the root symbol tells us something about your roots, your factors, right? For quadratic functions, for the quadratic formula, there's three things that can happen with b squared minus 4ac. We're going to lay it out here and we're going to talk a lot more about this, but just to give you a teaser, it basically looks like this. Now if the discriminant, I'm just going to use a symbol, dude, there are other symbols that people use, right? Let's use it all around. One of them is, I think it's the triangle. But anyway, we're just going to use d as representing the discriminant, right? So if the discriminant is greater than zero, basically it means it's a positive number, right? You can take the square root of a positive number and as we talked about, the square root of anything is plus or minus. That's where the plus and minus is coming from in the quadratic formula, right? So if d is greater than zero, then we're going to have a plus or minus, so there's going to be two real roots, right? Because we're going to have x is equal to negative b plus the square root of whatever that number is, divided by 2a, and x is equal to negative b, right? So that's two different roots, right? And those are two different x-intercepts. So it's telling us that the function, the parabola, crosses the x-axis twice. If d, if d, the discriminant is equal to zero, so if inside the root symbol here is zero, then what happens? The square root of zero is zero, so all we're going to have is one real root. There are basically, you know, there's different terminology for this. They call it two identical roots, one real root, right? So what happens is, as soon as d is equal to zero, the discriminant inside the root symbol is equal to zero, then all you get is x is equal to negative b over 2a, so that means your parabola crosses the x-axis or touches the x-axis, it doesn't cross it, it touches the x-axis once. So if you have your Cartesian coordinate system like this, the parabola comes down, touches it and goes up, or parabola comes up and touches it and goes up, right? When d is greater than zero, the parabola crosses the x-axis or goes down like this, right? If the discriminant is less than zero, if the discriminant is less than zero, it means it's negative, right? Anything less than zero is negative, and you can't take the square root of a negative number in the real number realm where we're working right now, right? You can't take the square root of a negative number, then what happens is your parabola doesn't touch or cross the x-axis, so if you have your Cartesian coordinate system, your parabola either is formed above it or is formed below it and it doesn't hit the x-axis, right? And this is called the discriminant, it gives us a lot of information about the function and, you know, the nature of the function, the nature of the roots, basically, right? And we're going to talk a lot more about this when we start graphing polynomials. Right now what we're concerned about is using the formula to solve quadratic functions or to factor quadratic equations, right, or anything in this form. Basically, and we're going to do one like this. We're going to do two or three questions, examples like this, and we're going to do one like this. And later on when we get into factoring polynomials, we're actually going to drive this formula. And this formula is just basically, all it is is messing around with the equal sign and there's something called completing the square. So what we're going to do is use completing the square and drive this formula when we get into graphing polynomials, graphing quadratic functions, which is super important, right? It's a huge section in my area anyway, in grade 11 mathematics. And, you know, we're going to touch on this right now. We're just going to, you know, solve equations, find the roots, find the factors, find the x intercepts. And then once we've learned all our different factoring techniques, we're going to go into the polynomial section and start graphing some of these polynomials.