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in this video I'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics and if you've seen many of my videos you know that I'm not a big fan of memorizing things but I will recommend you memorize it with the caveat that you also remember how to prove it because I don't want you just to remember things and and not know where they came from but with that said let me show you what I'm talking about it's the quadratic quadratic formula and as you might guess it is to solve for the roots or the zeros of quadratic equations so let's speak in very general terms then I'll show you some examples so let's say I have an equation of the form ax squared plus BX plus C is equal to zero you should recognize this this is a quadratic equation where a B and C are well a is the coefficient on the x squared term or the second degree term B is the coefficient on the X term and then C is you can imagine the coefficient on the X to the zero term where it's the constant term now given that you have a general quadratic equation like this the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac all of that over 2a I know it seems crazy and convoluted and hard for you to memorize right now but as you get a lot more practice you'll see that it actually is a pretty reasonable formula to stick in your brain someplace and you might say gee this is a wacky formula where did it come from and in the next video I'm going to show you where it came from but I want to just get from the get used to using it first but it really just came from completing the square on this equation right there if you complete the square here you're actually going to get this solution and that is the quadratic formula right there so let's apply it to some problems so let's start off with something that we could have factored just to verify that it's giving us the same answer so let's say we have x squared plus 4x minus 21 is equal to 0 so in this situation a is equal to let me do that in a different color in this situation a is equal to 1 right the coefficient on the x squared term is 1 B is equal to a 4 B is equal to 4 the coefficient on the X term and then C is equal to negative 21 the constant term and let's just plug it in the formula so what do we get we get X this tells us that X is going to be equal to negative B negative B is negative 4 negative 4 I put a negative sign in front of that negative B plus or minus the square root of B squared B squared is 16 all right 4 squared is 16 minus 4 times a which is 1 times C which is negative 21 so we can put a 21 out there and that negative sign will cancel out just like that with that but actually let me this is the first time we're doing it let me not skip too many steps so negative 21 just so you can see how it fit in and then all of that all of that over 2 times a over 2 times a a is 1 so all of that over 2 so what does this simplify or hopefully it simplifies so we get X is equal to negative 4 plus or minus the square root of C we have a negative times a negative that's going to give us a positive and we have 16 plus let's see this is 6 4 times 1 is 4 times 21 is 84 16 plus 84 is 100 that's nice it's a nice perfect square all of that over 2 and so this is going to be equal to negative 4 plus or minus 10 over 2 we could just divide both of these terms by 2 right now so this is equal to negative 4 divided by 2 is negative to plus or minus 10 divided by 2 is 5 so that tells us that X X could be equal to negative 2 plus 5 which is 3 or X could be equal to negative 2 minus 5 which is negative 7 so the quadratic formula seems to have given us an answer for this you can verify just by substituting back in that these do work or you could even just try to factor this right here you say what two numbers when you take their product you get negative 21 and when you take their sum you get positive 4 well that's so you'd get X plus 7 times X minus 3 is equal to negative 21 notice 7 times negative 3 is negative 21 7 minus 3 is positive 4 you would get X plus sorry it's not negative 21 is equal to 0 there should just be a 0 there so you get X plus 7 is equal to 0 or X minus 3 is equal to 0 X could be equal to negative 7 or X could be equal to 3 so it definitely gives us the same answer as factoring so you might say hey why bother with this crazy mess and the reason we want to bother with this crazy mess is it'll also work for problems that are hard to factor and let's do a couple of those let's do some hard to factor problems right now so let's scroll down get some fresh real estate let's rewrite the formula again just so we have in case we haven't had it memorized yet X is going to be equal to negative b plus or minus the square root of b squared minus 4ac all of that over 2a now let's apply this to another problem let's say we have let's say we have the equation 3x squared plus 6x is equal to negative 10 well the first thing we want to do is get it in the form where all of our terms are on the left hand side so let's add 10 to both sides of this equation we get 3x squared plus six X plus 10 is equal to zero and now we can use the quadratic formula so let's apply it here so a is equal to three that is a this is B and this right there is C so the quadratic formula tells us the solutions to this equation the roots of this of this of this quadratic function I guess we could call it X is going to be equal to negative B negative B B is 6 so negative 6 plus or minus the square root of B squared B is 6 so we get 6 squared minus 4 times a which is 3 times C which is 10 stretch out the radical a little bit all of that over 2 times a 2 times a 2 times 3 so we get X is equal to negative 6 plus or minus the square root of 36 36 minus this is interesting minus 4 times 3 times 10 so this is minus this is minus let me make sure yeah 4 times 3 times 2 is minus on 120 minus 120 all of that over all of that over 6 so this is interesting you might already realize why it's interesting what is this going to simplify to 36 minus 120 is what that's 84 if I'm doing my limit 120 minus 36 we make this into a 10 then this will become an 11 this is a 4 it is 84 so this is going to be equal to negative 6 plus or minus the square root of but not positive 84 that's if it's 120 minus 36 we have 36 minus 120 it's going to be negative 84 negative 84 all of that all of that over 6 so you might say gee this is crazy weather use a silly quadratic formula you're introducing me to Sal it's worthless it just give me a square root of a negative number it's not giving me an answer and the reason why it's not giving you an answer at least an answer that you might want is because this will have no real solutions no real solutions in the future we're going to introduce something called an imaginary number which is a square root of a negative number and then we can actually express this in terms of those numbers so this actually does have solutions but they involve imaginary numbers so this actually has no real solution we're taking the square root of a negative number so the B squared with the b squared minus 4ac if this term right here is negative then you're not going to have any real solutions and let's verify that for ourselves let's get our graphing calculator out let's graph this let's graph this equation right here so let's get the graph the Y is equal to that's what I had there before so you have 3 X let me clear this right so I get 3x squared plus 6 X plus 10 so that's the equation we're going to see where it intersects the x axis where does it equal 0 so let me graph it let's graph it notice this thing just comes down and then goes back up its vertex is sitting here above the x-axis and it's upward-opening it never intersects the x-axis so at no point will this expression will this function equals 0 at no point will y equals 0 on this graph so once again the quadratic formula seems to be working let's do one more example you can never see enough examples here and I want to do ones that are you know maybe not maybe not so obvious to factor so let's say we get let's say negative 3x squared plus 12x plus 1 is equal to 0 now let's try to do it just having the quadratic formula in our brain so the X the X's that satisfy this equation are going to be negative B this is B so negative B is negative 12 plus or minus the square root of B squared of 144 that's B squared minus 4 times a which is negative 3 times C which is 1 all of that all of that over 2 times a over 2 times negative 3 so all of that over negative 6 this is going to be equal to negative 12 plus or minus the square root of what is this the negative times a negative they cancel out so I have 144 plus 12 so that is one 156 right 144 plus 12 all of that all of that over negative 6 now I suspect we can simplify this 156 we can maybe bring some things out of the radical sign so let's attempt to do that let's attempt to do that so let's do a prime factorization of 156 sometimes this is the hardest part simplifying the radical so 156 is the same thing as 2 times 78 78 is the same thing as 2 times what that's 2 times is that 2 times 39 2 times 39 so the square root of 156 so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that's the square root of 2 times 2 times the square root of 39 which and this obviously is just going to be square root of 4 or this is the square root of 2 times 2 is just 2 2 square roots of 39 if I did that properly let's see 4 times 39 yeah it looks like it's right 120 yep so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 all of that over negative six now we can divide the numerator and the denominator my BB by 2 so this will be equal to negative six plus or minus the square root of 39 over negative 3 or we could separate these two terms out we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 now this is just a 2 right here right these cancel out 6 divided by 3 is 2 so we get 2 and now notice if this is plus and we use this minus sign the plus will become negative and the negative will become positive but it still doesn't matter right we could say minus or plus or that's the same thing as plus or minus the square root of 39 over 3 I think that's about as simple as we can get this answer now I want to make a very clear point of what I did at that last step I did not forget about this negative sign I just said it doesn't matter it's going to turn the positive into the negative it's going to turn the negative into the positive let me rewrite this so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 right that's what the plus or minus means it could be this or that or both of them really now in this situation this negative 3 will turn into 2 minus the square root of 39 over 3 right I'm just taking this negative out here the negative and the negative will become a positive and you get 2 plus the square root of 39 over 3 right negative times a negative is a positive so once again you have to plus or minus the square root of 39 over 3 2 plus or minus the square root of 39 over 3 our solutions are solutions to this to this equation right there let's verify I'm just curious what the graph looks like so let's just look at it so let's look let me clear this where's the Clear button so we have negative 3x squared plus 12x plus one and let's graph it let's see where it intersects the x-axis goes up there and then back down again and then so what are the so two plus or minus the square C's square root of 39 square root of 39 it's going to be a little bit more than six right because squit 36 is 6 squares there's me a little bit more than 6 so this is going to be a little bit more than 2 a little bit more than 6 divided by 3 is a little bit more than 2 so you're going to get one value that's a little bit more than 4 and then another value that should be a little bit less than one and that looks like the case you have 1 2 3 4 you have a value that's pretty close to 4 and then you have another value that is a little bit a little bit maybe it looks close to 0 but a little bit less than that so anyway hopefully you found this application of the quadratic formula helpful