The quadratic formula | Algebra (video) | Khan Academy

Video transcript

In this video, I ‘m going to expose you to what is possibly one of at least the top five most useful formulas in mathematics. And if you ‘ve seen many of my video, you know that I ‘m not a bad fan of memorizing things. But I will recommend you memorize it with the caveat that you besides remember how to prove it, because I do n’t want you to just remember things and not know where they came from. But with that said, let me show you what I ‘m talking about : it ‘s the quadratic formula. And as you might guess, it is to solve for the roots, or the zeroes of quadratic equations. So let ‘s talk in very general terms and I ‘ll show you some examples. So let ‘s say I have an equality of the form ax squared plus bx plus hundred is equal to 0. You should recognize this. This is a quadratic equation where a, barn and carbon are — Well, a is the coefficient on the adam squared term or the second degree terminus, boron is the coefficient on the adam term and then c, is, you could imagine, the coefficient on the ten to the zero condition, or it’s the changeless term. now, given that you have a general quadratic equation like this, the quadratic recipe tells us that the solutions to this equation are x is equal to negative b plus or minus the squarely root of bacillus squared minus 4ac, all of that over 2a. And I know it seems crazy and byzantine and difficult for you to memorize right now, but as you get a draw more practice you ‘ll see that it actually is a pretty reasonable rule to stick in your brain somewhere. And you might say, gee, this is a balmy recipe, where did it come from ? And in the following video I’m going to show you where it came from. But I want you to get used to using it first. But it truly equitable came from completing the square on this equation right field 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. Let ‘s start off with something that we could have factored merely to verify that it ‘s giving us the lapp answer. So let ‘s say we have x squared plus 4x minus 21 is equal to 0. so in this position — let me do that in a different discolor — a is equal to 1, right ? The coefficient on the adam square condition is 1. b is equal to 4, the coefficient on the x-term. And then c is equal to damaging 21, the constant term. And let ‘s just plug it in the formula, so what do we get ? We get ten, this tells us that ten is going to be equal to negative b. negative boron is negative 4 — I put the veto sign of the zodiac in front of that — veto b plus or minus the public square rout of boron squared. boron squared is 16, 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 — Since this is the beginning time we ‘re doing it, let me not skip besides many steps. therefore negative 21, just so you can see how it fit in, and then all of that over 2a. 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 — Let ‘s see we have a negative times a damaging, that ‘s going to give us a positive. And we had 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. That ‘s a decent perfect public square. All of that over 2, and therefore this is going to be peer to damaging 4 plus or minus 10 over 2. We could precisely divide both of these terms by 2 right now. So this is peer to veto 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5. So that tells us that adam could be equal to damaging 2 plus 5, which is 3, or ten could be equal to damaging 2 subtraction 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 evening precisely 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 ? so you ‘d get adam plus 7 times ten subtraction 3 is equal to negative 21. Notice 7 times negative 3 is damaging 21, 7 subtraction 3 is positive 4. You would get ten plus — sorry it ‘s not negative — 21 is equal to 0. There should be a 0 there. So you get x plus 7 is adequate to 0, or x minus 3 is adequate to 0. ten could be adequate to negative 7 or x could be equal to 3. So it decidedly gives us the lapp answer as factorization, so you might say, hey why annoy with this crazy fix ? And the rationality we want to bother with this crazy mess is it ‘ll besides 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 permit ‘s scroll down to get some clean veridical estate. Let ‘s rewrite the formula again, precisely in encase we have n’t had it memorized yet. ten is going to be equal to negative b-complex vitamin plus or minus the squarely solution of bacillus squared minus 4ac, all of that over 2a. I ‘ll supply this to another problem. Let ‘s say we have the equality 3x squared plus 6x is equal to negative 10. well, the inaugural thing we want to do is get it in the kind where all of our terms or on the left side, so let ‘s add 10 to both sides of this equality. We get 3x square plus the 6x plus 10 is adequate to 0. And nowadays we can use a quadratic convention. So let ‘s apply it here. So a is equal to 3. That is a, this is barn and this right here is c. thus the quadratic recipe tells us the solutions to this equation. The roots of this quadratic function, I guess we could call it. x is going to be equal to negative b. b is 6, thus negative 6 plus or minus the square root of bel squared. barn is 6, so we get 6 squared minus 4 times a, which is 3 times c, which is 10. Let ‘s stretch out the extremist little act, all of that over 2 times a, 2 times 3. So we get ten is equal to negative 6 plus or minus the square rout of 36 minus — this is concern — minus 4 times 3 times 10. so this is minus — 4 times 3 times 10. so this is minus 120. All of that over 6. so this is concern, you might already realize why it ‘s interest. What is this going to simplify to ? 36 minus 120 is what ? That ‘s 84. We make this into a 10, this will become an 11, this is a 4. It is 84, indeed this is going to be equal to veto 6 plus or minus the squarely root of — But not positive 84, that ‘s if it ‘s 120 subtraction 36. We have 36 minus 120. It ‘s going to be negative 84 all of that 6. so you might say, g, this is crazy. What a this airheaded quadratic formula you ‘re introducing me to, Sal ? It ‘s despicable. It just gives me a feather 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. In the future, we ‘re going to introduce something called an complex number number, which is a feather 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 fanciful numbers. indeed this actually has no actual solutions, we ‘re taking the square root of a negative number. So the barn squared with the b squared minus 4ac, if this term good here is minus, then you ‘re not going to have any veridical solutions. And let ‘s verify that for ourselves. Let ‘s get our graphic calculator out and let ‘s graph this equality right hera. then, let ‘s get the graph that y is equal to — that ‘s what I had there ahead — 3x squared plus 6x plus 10. So that ‘s the equality and we’re going to see where it intersects the x-axis. Where does it equal 0 ? therefore let me graph it. Notice, this matter precisely comes depressed 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. therefore at no orient will this saying, will this serve, equal 0. At no distributor point will y equal 0 on this graph. so once again, the quadratic convention seems to be working. Let ‘s do one more case, you can never see adequate examples here. And I want to do ones that are, you know, possibly not so obvious to agent. So let ‘s say we get negative 3x squared plus 12x plus 1 is equal to 0. nowadays let ‘s try to do it just having the quadratic formula in our brain. So the adam ‘s that satisfy this equation are going to be negative barn. This is b so negative b is negative 12 plus or minus the square root of boron squared, of 144, that ‘s boron squared minus 4 times a, which is negative 3 times c, which is 1, all of that over 2 times a, over 2 times damaging 3. So all of that over negative 6, this is going to be equal to negative 12 plus or minus the squarely settle of — What is this ? It ‘s a negative times a negative so they cancel out. so I have 144 plus 12, so that is 156, right ? 144 plus 12, all of that over negative 6. immediately, I suspect we can simplify this 156. We could possibly bring some things out of the radical bless. So let ‘s attack to do that. So lashkar-e-taiba ‘s do a prime factorization of 156. sometimes, this is the hardest character, simplifying the radical. so 156 is the like thing as 2 times 78. 78 is the same matter as 2 times what ? That ‘s 2 times 39. So the square root of 156 is equal to the square solution of 2 times 2 times 39 or we could say that ‘s the hearty root of 2 times 2 times the square solution of 39. And this, obviously, is merely going to be the square root of 4 or this is the square settle of 2 times 2 is barely 2. 2 square roots of 39, if I did that by rights, let ‘s see, 4 times 39. Yeah, it looks like it ‘s veracious. so this up here will simplify to damaging 12 plus or minus 2 times the square solution of 39, all of that over damaging 6. now we can divide the numerator and the denominator possibly by 2. sol this will be equal to damaging 6 plus or minus the squarely etymon of 39 over veto 3. Or we could separate these two terms out. We could say this is peer to negative 6 over veto 3 plus or minus the squarely 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 subtraction signal, the plus will become minus and the damaging will become plus. But it still doesn’t topic, good ? We could say minus or plus, that ‘s the same thing as plus or minus the square root of 39 nine over 3. I think that ‘s approximately a elementary as we can get this answer. I want to make a very clear point of what I did that last measure. I did not forget about this negative sign. I just said it does n’t matter. It ‘s going to turn the incontrovertible into the negative ; it ‘s going to turn the negative into the positive. Let me rewrite this. thus this properly here can be rewritten as 2 plus the square solution 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, actually. now in this situation, this negative 3 will turn into 2 minus the square root of 39 over 3, right ? I ‘m barely 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 ? A negative times a negative is a positive. thus once again, you have 2 plus or minus the square of 39 over 3. 2 plus or minus the hearty settle of 39 over 3 are solutions to this equation right there. Let affirm. I ‘m barely curious what the graph looks like. So let ‘s just look at it. Let me net this. Where is the clear button ? So we have negative 3 three squared plus 12x plus 1 and let ‘s graph it. Let ‘s see where it intersects the x-axis. It goes up there and then second gloomy again. so 2 plus or minus the square, you see — The feather etymon of 39 is going to be a little sting more than 6, right ? Because 36 is 6 square. So it ‘s going be a little bite more than 6, so this is going to be a fiddling bite more than 2. A little bite more than 6 divided by 2 is a little bit more than 2. So you ‘re going to get one value that ‘s a little moment more than 4 and then another value that should be a little snatch less than 1. And that looks like the lawsuit, you have 1, 2, 3, 4. You have a measure that ‘s pretty conclude to 4, and then you have another value that is a little morsel — It looks close to 0 but possibly a small bit less than that. thus anyhow, hopefully you found this application of the quadratic equation recipe helpful.

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