Okay. So now that we've reviewed lines, some of the other common equations that you'll get in physics will be quadratic equations. Remember, a quadratic is when you have something like a variable like x raised to the second power instead of the first power. So this is an equation in a standard form of a quadratic. There are basically just two different ways that we're going to use in physics to solve. We're going to take square roots, or we're just going to use the quadratic formula. And it really just depends on how the quadratic looks. So, for example, if you have something like x+2 equals 4, that looks like where you have x+number that's squared, and then you have a constant on the other side. If you have something that looks like this in a little bit more of a standard form, that kind of looks like ax2+bx+c, then you could always just use the quadratic formula, but it's a little bit more tedious. I'm actually going to show you that these two things are the exact same equation written in different ways. We're going to get the same exact solutions for this. So let me show you how to take the square root. Basically, the way this works is if you have something looks like this, in order to get x by itself, I'm going to have to get rid of the squared over here, so I'm going to have to isolate the squared expression, and then I have to take the square root of both sides. And when I do that, I'm going to have to take the positive and negative roots. That's always how you take square roots. So what ends up happening is the squared and square root cancel, and you're left with x+1=±4. So what this works out to is you have x=-1±2. And so what this is saying is that there are two solutions. There's negative 1 plus 2, which ends up being just 1, and there's negative one minus 2, which ends up being negative 3. So those are your two solutions to this quadratic. Usually, quadratics will give you two solutions.

Now, that's how to take square roots, but if you're ever given a quadratic that looks like this in which you can't factor it or you don't know what method to use, you can always pop it into the quadratic formula that will always give you your correct answers. Alright? And this is just a little bit more tedious, but basically, just remember here that the coefficients of your quadratic are a, b, and c, and you just pop the appropriate expressions and letters into this formula over here. So in other words, this is going to be x=-b±b2-4ac/2a, which in this case is just 2 times 1. So just to simplify a little bit, this is going to be x=2±4--12/2. And if you simplify this even further, what this turns out to be is it turns out to be x=-1±2, which is exactly what we got over here. Remember, this just means two things. It means negative 1 plus 2, which in this case is 1, negative 1 minus 2, which in this case is negative 3. So it makes sense that we got the exact same answers whether we use the square root or the quadratic, because, again, these two equations are the exact same thing. You can just check that if you distribute and sort of foil this out and convert it to standard form, these things are the same.

Alright? So that's how to use quadratics and square roots to solve quadratics. Alright. So now that we know how to solve quadratics algebraically, a lot of times you're going to see quadratics graphed, and you'll have to identify some key information from graphed quadratics. Remember that quadratics kind of look like parabolas. They can either look like smiley faces or frowny faces, and it just depends on their equation. Now the equation that you'll see most of the time for quadratics is going to be in vertex form because it's easy to plot them. And basically, it looks like a(x-h)2+k. The vertex, meaning the little points where the parabola opens up, is going to be the coordinate h,k. So in this case, we have -x-12+4. The coordinate of the vertex is actually just going to be 1,4. And because of this negative sign, it's actually going to look like a frowny face. So what we can see here is that the coordinate 1,4 is going to be the sort of top of our parabola. It's going to be the maximum point over here, and it's basically going to go, like, looking like this. In order to get those points, we're going to have to solve for the x and y intercepts. And basically, to do that, you're just going to solve for when y is equal to 0 for the x. And then for the y intercept, you're going to set x equal to 0 and solve for that variable over there. So I'm just going to actually go ahead and do this for you. It's actually just the of the quadratics that we solved, up above. And so what this is going to look like, it's going to be like negative 13. That's the point where it crosses the x axis, so it's going to look something like this. Whereas the y intercepts are where you set the x coordinate equal to 0, so you basically just set this equal to 0 and solve. What you're going to see is that this quadratic hits the y intercept or the y axis at the point 3. So once you solved all your points, you're going to connect them with a smooth curve just to see what your parabola looks like. We basically just connect all these things over here with a little curve, and so our parabola looks like a little frowny face that looks like that. And from this graph, we can tell when the graph is increasing and decreasing. Those are going to be important things. So, for example, we can see here that as we're moving left to right, the graph is increasing up until we get to the x equals one point. So it's increasing when x is less than 1. And then as you're going left to right, it starts decreasing everywhere after positive one. Alright? So that's how to graph and understand key information from quadratic equations.