 Welcome back and in this video I'm going to talk about quadratic or how to solve equations using the quadratic formula It's a very popular method of solving quadratic equations. So here we go now in the middle of screen What I have here is the actual quadratic formula about read this first if a quadratic equation is in standard form So that means if I have ax squared plus bx plus c So the x squared is first then I have to be then I have to see if set equal to zero Then the solutions or the roots do think I want to say it are going to be here So this this formula here gives us Basically what I have to do is plug in the a's Here the B's here and the C's right here I only use it once use the C right there and if I plug those numbers in and then add subtracts multiply divide do all That's things correctly. I can find the solutions for any quadratic equation Okay, now. This is a very handy formula to use because if I have a very complicated equation then I just plug in the numbers Figure out what it is. It's relatively straightforward. Yeah, so I'm going to do two examples for this The first example that I have here is quadratic functions with real zeros Now what that basically means is there's real numbers and then there's imaginary numbers or real numbers and complex numbers Okay, so in mathematics. We have different distinctions for numbers We have different categories for numbers in this one. We're just going to get some real solutions We're gonna get some real numbers when we solve this Okay, so the first thing that I want to do is I want to set this equal to zero now As you can already tell it is kind of already say equal zero it's a equal to f of x But I can I can rewrite it to x squared minus 16 x plus 27 equals zero Now the reason that you do that is just to make sure everything is in order Everything goes where it's supposed to be you got your x squareds here Then your x is here and then your constants there So we have the a number the b number the c numbers. Let me label those we have the a number here We have the b number here. We have the c number here now those that a b and c number We're going to use that I'm going to plug that into the quadratic formula for our to solve Okay, so over here. I'm going to write the quadratic formula x equals now This is something you should just commit to memory. It's such a useful formula You're using a lot of your different math classes So this is something you just should commit to memory Okay, x equals negative b plus or minus the square root of b squared minus 4 a c all Over 2 a okay again just a handy little formula to memorize Okay, depending on your teachers Sometimes they let you have no card. Sometimes they write the formulas on the board Sometimes they give you a sheet whatever the case is again. This is the one that you should just memorize It's so useful in everything that you do. Okay, especially in your future classes So anyway, I'm going to plug in the a's b's and c's just plug them in so in this case I have negative b so this is actually a negative negative 16 So that's actually going to make it positive. I'll still put it in there But it's actually going to make that 16 positive. Okay, but you still want to plug it in there So you can see why it's a positive 16 there A negative negative 16 plus or minus the big square root of negative 16 squared Okay, now it's not quite as Important to put in a negative there because then even if that number is negative You're going to square it to get a positive number. So you don't necessarily need it all the time four times a a in this case is 2 and Times c c in this case is 27 Okay, and then all over Two times a two times two which is eventually going to be four Okay, so now what we're going to do is we got to do just all the rest of this is basically just arithmetic Okay, so now x is equal to Positive 16 plus or minus the big square root. See if I can do some mental math here 16 squared is 256 256 now it's really gonna test me here. This is four times two which is eight and then eight times 27 eight times 20 is Gonna be a hundred and sixty hundred and sixty and Then yeah eight times 20 is gonna be a hundred and sixty and then eight times seven is fifty six 56 so that's going to be 216 so minus two hundred and sixteen all divided by two times two which I said earlier was four Okay So now I'm gonna do a little bit of racing here to make this kind of nice and nice and neat So as you go over this and notice that some of these numbers are gonna be nice and neat So I got 16. I got a four 256 minus 16. That's simply just going to be 40 40. Yep so x is Equal to 16 plus or minus the square root of 40 all over four now I said that this this was nice and neat it kind of is but the square root of 40 We actually don't know what that is so we actually got to simplify that just a little bit So it's not gonna be as nice neat as I said it was kind of fit there But we will be able to get some exact answers here We won't get numbers like five and negative 17 or something like that But we will get an exact answer here But what we need to do first is we need to reduce this square root of 40 actually I'm running out of room down here So I'm actually going to move everything up here Okay, so bear with me on the space that I have so what I'm going to do now is x equals 16 now 40 is 4 times 10 So I'm going to reduce that to plus or minus 2 root 10 Okay, so 40 is this you can also reduce or split that up into the square root of 4 and the square root of 10 Now the square root of 4 is 2 square root of 10. We just leave there Okay, and then divided by 4 Okay, now now as you look at this you might think oh, okay, I might be done or you might also think oh Hey, I can I can reduce some numbers here, right now be very careful with this. This is a mistake That's that a lot of students make here is that they'll just They'll just divide the 16 and the 4 and they'll totally forget about this 2 over here If you're going to reduce you got to reduce everything if you're going to divide you got to divide everything So it's going to be 16 divided by 4 and 2 divided by 4 So there's a couple different ways to do this you can actually just divide those so take x equals 16 divided by 4 is 4 and Then 2 divided by 4 is going to be one half So that's one way to write this now This is not a form that you will probably see a lot of your teachers or a lot of books any answers from books You might see they probably won't write it like this because a lot of times they don't like fractions like this Instead what they how they could write this how they could write this is? Instead of just taking everything divided by 4 what they'll do is they'll reduce by some common factor So what they'll do is they'll just reduce by some common number So 16 4 and 2 all these numbers are divisible by 2 so I'm just going to divide by 2 divide by 2 and Divide by 2 ok so that this right here is going to be the most common way of Writing your answer ok that's going to be the most common way of writing your answer Now this bottom one is still technically correct But that one you won't see very often simply because well it's a little bit messy with that one half on the outside there So that's usually not how it's written most of the time. We like to write these as single fractions. Here's my numerator Here's my denominator. That's how we usually write that ok, but now as I said before those are two real numbers Those are two real numbers Two real solutions that we're going to get now again. You can see why it's two numbers eight plus the square root of 10 divided by two and Eight minus the square root of 10 divided by two that's going to give us two different solutions there ok So that's that's a quadratic functions That's using it to get real solutions now I'm going to do another example where we actually get complex solutions where we actually get We're going to get some imaginary numbers here Okay, so same directions find the zeros of the following function using the quadratic formula Okay, so the first thing I'm going to do is I want to write down the quadratic formula x equals negative b plus or minus the big square root of big b squared minus 4 a c all of this Over to a again. This is something you should just memorize So my a number in this case now I don't have to rearrange it So I'm not going to write it like I did last time my a number is 4 my b number is 3 and my c number is 2 Okay, so I'm just going to start solving this all right So I'm just gonna start plugging in numbers x equals negative 3 plus or minus the big square root of 3 squared minus 4 times 4 times 2 all over 2 times 4 okay, just plugging in my a's b's and c's as I was right now down I was looking back and forth back and forth left and right left and right figuring out Okay, my a number is 4 my b number is 3 my c numbers to I do look back and forth to find them Okay, so simplify this out a little bit negative 3 plus or minus the big square root of Okay, let me not make that big okay big square roots of 3 squared is 9 minus 4 times 4 which is 16 16 times 2 is 32 Okay, now for those of you who know about imaginary solutions there it is you can automatically see right there We're gonna get imaginary solutions 9 minus 32 is going to be a negative 23 Negative numbers underneath the square roots gives me imaginary solutions so you could automatically see right here and now I'm gonna get imaginary solutions in this case complex solutions Okay, two times four on the bottom here is going to be eight okay now I'm gonna keep going through this keep doing the process okay keep simplifying okay x equals negative 3 plus or minus the Big square root of negative 23 all over 8 now on the previous example We were able to simplify we were able to reduce some numbers in this case. Nope We're not able to reduce anything Now there's 23 over here again, that's that's that's not going to reduce or anything like that But the negative inside kind of will okay, so I'm actually going to go over here with my final answer x is equal to negative 3 plus or minus The square root of a negative number is an imaginary number So what that what that basically does is you take the negative part you bring it out You make it imaginary you make it this this I kind of look variable looking thing Okay, so it's gonna be negative 3 negative 3 plus or minus I root 23 all over 8 and that is the solution You're most commonly going to see another way that you could write this It's not it just looks a little bit different. It's not wrong. It just looks a little bit different. I you can actually take You can actually divide everything by 8 so you get negative negative 3 eighths here and then I Wrote 23 over 8 you might see the solution like that But that was not more more commonly used Simply just because it's messy. It's got a bunch of fractions in it. It's just messy. This one is nice cleaner It's nicer. Everything's kind of one fraction got your numerator up here You denominator down here. This is kind of easier to work with then then two fractions down here But anyway, all right, so that is those those your complex is complex zeros You got some real numbers like negative 3 and 8 in here and then you go to your imaginary numbers this I wrote 23 Okay, so you got some real numbers in there and imaginary numbers. That's what makes a complex solution Okay, all right. I think that's it for this video. I hope you enjoyed this video of going over quadratic formula It's it's one of the better formulas to use and again, please just just memorize it makes life a lot easier Makes your math life a lot easier. Yeah, I hope you enjoyed this video Hope you learned something today, and we'll see you next time