Examples: A Different Way to Solve Quadratic Equations

hi I'm potion lo I've been thinking about how to explain math and more thoughtful and interesting ways a few months ago I was thinking about quadratic equations and I was really surprised to find out that you can solve them in this really simple way I couldn't believe that I had never seen this before in any textbook so I'm sharing a detailed explanation of the method here with examples first I'll show the method for a few seconds just jump back here in the video and pause whenever you want to see it again the words are carefully chosen so that the method is logically sound let's get started this technique actually doesn't require much more background knowledge than what's called the distributive property to review that suppose I have X minus 3 multiplied by X minus 4 how do you expand these brackets just remember you find all the ways of taking something from each of the brackets and multiplying together when I do that if I take the X and the X I get x squared if I have the X and the minus 4 I have minus 4x if I take the minus 3 and the X I have minus 3x and if I take the minus 3 and the minus 4 they multiply to 12 and usually we simplify this expression by putting together the terms that are similar meaning that the same number of X's multiplied together in them so we can often write this as x squared minus 7x plus 12 before we go on I just want to look at this and notice a pattern where does this minus 7 come from that actually comes from the minus 3 plus the minus 4 I could also say where does the 7 come from it comes from the three plus the four if both of those have negative sites how about the twelfth it's the negative three times the negative four also if I was going to write down two numbers after negative signs their product would be that twelve because the two minus signs would multiply to make a plus let's review one more suppose I want to expand out a different bracket what if I have something like three minus u times three plus u here I just want to emphasize there's nothing magic about X algebra lets you use any letter that you want to use if I expand this out all the possible ways of combining pairs if I take the three and the three I get nine then the first three and the last you gives me plus three you the first minus U and the second three gives me minus three you and the last term is minus nu times plus u which is minus u squared so here if I want to simplify I could actually cancel the three u minus 3u and I'll get nine minus u squared actually that's kind of cute there is not as much as in this one this this expression has more stuff in the middle and that's because these two middle terms cancel and that's actually because of the form of what I have here because this three went without you and this negative you went with that three that's kind of neat and in math we actually love to see these kinds of cute coincidences they can sometimes come in handy later okay so this was a review but why would anyone actually care about this well first of all is useful to be able to expand brackets but second of all it turns out that this is an interesting way of solving quadratic equations let me write down this top piece again in a slightly different order you see I now know that I have another way to write x squared minus 7x plus 12 it's like that so x squared minus 7x plus 12 is also X minus 3 times X minus 4 now when we talk about a quadratic equation often times what's called a standard form quadratic equation is something like this thing on the Left equals zero you want to know when this thing is zero but there's something nice that happens if you can write this as a product of two things if I want to know exactly when is this whole thing zero I can just ask exactly when is this whole thing zero this is a product of two numbers if I have an X X as a number if you plug X in you'll get a number times another number if you multiply two numbers together and get zero that happens exactly when one or both of the numbers is zero that's actually called the zero product property and so if I know that this quadratic can be rewritten in this way that means that the quadratic is zero precisely when either this is 0 or that is 0 it's a precisely 1 so then when is X minus 3 equal to 0 X minus 3 is 0 exactly when x equals 3 on the other hand X minus 4 equals 0 precisely when x equals 4 so what this tells me is that the two solutions to this quadratic equation this equals 0 are exactly x equals 3 and x equals 4 those are the only values of X which if I put into here spit out 0 and also because if I need this to be 0 the only way I can multiply two numbers to get 0 is if one of them is 0 I spent a lot of time talking about this because the logic is actually quite important but the conclusion here is that if I'm trying to solve a quadratic equation if only I could find this magical factoring then I can read off the answers the answers are whatever make this 0 or that 0 in this case it's the 3 and the 4 what I've talked about so far is actually not a new technique it's just called the method of factoring for solving quadratic equations the the factoring is to try to take your quadratic and write it as a product like this let's try using this factoring to solve a question x squared minus 14x plus 24 equals 0 now if we want to do this by factoring we would like to try to write x squared minus 14x plus 24 we would like to write it as X minus something times X minus something else sometimes in books you'll see plus signs instead of minus signs here I'm using minus signs on purpose because that's going to let us learn something deeper in math as we keep going and it's also going to be more convenient so I would really like to find two numbers to put there just as we saw before if I find these two numbers then those two would actually be the solutions of this quadratic equation and what do these two numbers need if you think about what happens as you expand the brackets I would need to find these two numbers so that they add to 14 and they multiply to 24 I didn't say add to minus 14 they have to add to positive 14 because I already have minus signs here I'm talking about what do I write in these pinkish-purple blanks I want to write two numbers there that add to positive 14 and multiply to positive 24 and I'll write that down like this if I can find two numbers with a sum equals to 14 and a product equals to 24 then those two numbers are going to be the solutions of this quadratic equation and not only the solutions they are all the solutions to the quadratic equation so the game then becomes to find these two numbers with this sum in this product notice there's a big word if here though because I don't actually know that these two numbers exist and the way that people usually go about finding these is actually with the method called guess and check that's what I learned as well when I was going to school but it turns out there's a much much easier way and this was the this was the thing that surprised me and I wanted to share with everybody else you see we're used to thinking of this as a factoring method so we always try to factor the product and we're thinking about all these different ways to prime factorize 24 and then find ways to take two factors of 24 that multiplied together they make 24 and add to 14 the trick is actually to start from the Sun I need to find two numbers with a sum of 14 and a product of 24 two numbers have a sum of 14 exactly when their average is seven and two numbers have an average of seven exactly when those two numbers are 7 minus a little bit and 7 plus the same bit so what I'll write here is an arrow and I'll say I just need to find a you so that seven - u + 7 + u give the product because if I found a u so that 7 - u + 7 + you gave that product they would already give the sum they're already set up so that it's the average - a bit and the average plus the same bit why did I call this u I want to use you for unknown I guess so now I just need to find these this u so that this happens but this is where the beautiful thing happens what is the product of 7 - u + 7 + u that's actually the beautiful form that we saw at the outset of this video if you multiply this together you would actually get 49 minus u squared because the middle terms are going to cancel that's the product of 7 - u + 7 + u i need that that gives me the product of 24 all of this in yellow is actually equivalent finding a u so that 7 minus u times 7 + u gives the product is exactly the same sentence as saying that 49 minus u squared equals 24 because that's just the product of those and that 24 is the product that we seek can we find a u so that this is true if you do a little bit of arithmetic if I found a u squared that was 49 - 24 that would work so actually I'm okay if I find a u squared which is 49 minus 24 which is 25 I'm gonna draw an arrow up and what that means is if I have a u squared which is 25 then I have that do you see a u which squares the 25 yes 5 so I can actually say u equals 5 gives me that u squared equals 25 and so on I'm going to emphasize that this exists that means a choice of you exists to make all of these happen which means that I can get two numbers with us 14 and a product of 24 just put five into there if you put five into there what do you actually get you get 7 minus 5 which is 2 and 7 plus 5 which is 12 and you can check that those two numbers do add to 14 and multiply this 24 but even if you didn't check we would know it had to work because all of these arrows flow upwards that's how the mathematical logic is working here you might be wondering why didn't I write plus or minus 5 it's because I actually don't need the other direction in order to finish I just need to know there's a flow from here up by the way there's also a flow if I chose to write a minus sign here instead if I said u equals minus 5 that also gives me that u squared is 25 and I can keep going and what I'll get from this is I would get that 7 minus negative 5 which is 12 and 7 plus negative 5 which is 2 are two numbers that work those are the same pair so that's just showing that whether you wrote the plus or the minus it actually wouldn't make a difference but because I found out that these two numbers exist that if is done and so I get 2 then and they're all the solutions and this is a technique that lets you do this without any guessing you might say it seems like I took forever to explain this technique that's because this one time I explained it I wanted to make sure that the flow of logic was 100% clear ok let's try some other equations x squared minus 8x plus 13 equals to 0 if you're familiar with the factoring method you'll actually recognize this as something that can't be factored normally because 13 is a prime number but let's try writing down what we would like to do with factoring we would like to be able to write down a factorization x squared minus 8x plus 13 equals x minus something times X minus something else these are two blanks and just as before what this means is if we can find two numbers with some equals to 8 and product equals to 13 then they are all the solutions and again this is a positive 8 because I'm putting numbers into these blanks that are after - sites ok and as we can see we have a lot of trouble finding numbers that multiply to 13 because 13 is a prime if I try 1 and 13 1 times 13 works but 1 plus 13 is not 8 but if we start from the sum instead it's great what would give me a sum of 8 well an average of 4 will give me a sum of 8 that means that I need to find a u so that 4 minus u + 4 + u give the product because they're already going to give the sum when I add them together and if I multiply these two together it's again going to have the middle terms cancel in that's exactly the same thing as 16 minus u squared being equal to 13 maybe you can see what you should choose in order to get 16 minus u squared to be 13 I'll be in good shape if u squared is 3 that's 16 minus 13 and for that I'll be in good shape if I choose you to be positive square root of 3 if I chose you to be negative square root of 3 that will also work pulling all the way back up here this tells me that if I use 4 minus the square root of 3 and 4 plus the square root of 3 these are two numbers that would give us some of it and a product of 13 this is where it's really convenient not to have to actually multiply them together to finish up we're simply using the fact that the logic is flowing upwards if I used you to be root 3 then u squared is 3 then 16 minus u squared is 13 then have the product and I'm good sometimes you write this in a little bit more compact way another way to write the answer is four plus or minus the square root of three and that's because it has both the minus and the plus inside it there's actually something quite interesting about this solution here because these answers are not integers so it actually be very very difficult to guess them but this shows that this technique actually can solve any quadratic equation even when the solutions aren't integers or rational numbers that you can easily guess and in fact this works for all kinds of quadratic equations let's now do a bunch of other ones that just show how general this technique is x squared minus 8x plus 18 equals to 0 by now hopefully it's familiar what we're doing so I'm just gonna say if can find two numbers with some equals 8 and product I'll write PR OD product equals 18 then they are all the solutions ok let's go faster to get a sum of 8 all I would need to use is two numbers which are equal distance from the average so I would just need a u so that 4 minus u + 4 + u to give the product of 18 and if I expand what happens when I multiply those together that's the same as saying 16 minus u squared is equal to 18 in order to get that I could achieve that by having u squared equal to 16 minus 18 in this case it's negative 2 can you have a number squared to be a negative number you can if you allow imaginary numbers complex numbers and I will say that's actually one of these reasons why complex numbers are so useful the fact that we can go ahead and say there is a choice of you that makes this true allows this technique to work but also allows you to solve quadratic equations in general you might be wondering is it okay to just go and add a number like this that doesn't seem to exist in ancient times people have the same misgivings the same worries about negative numbers like negative 3 if you ask somebody what is negative 3 they might say I'm not sure if you can show me negative 3 sheep I'll believe you but these days we seem to understand that negative numbers are useful for doing calculations because it's useful to be able to have your answer become negative and then add some stuff and have it be positive again the same thing is happening here I can get you squared to be minus 2 by choosing you to be the square root of minus 2 and the square root of minus 2 is the imaginary number I times the square root of 2i is the square root of minus 1 by choosing this particular you then its square is minus 2 and I can keep going and it works and if you put a minus sign in front that would also be ok but we don't need that here and so going back up here we now find out that two numbers that sum to 8 and multiply to 18 are 4 plus or minus I times the square root of 2 and so this technique even works when the solutions are imaginary when you see this you actually realize that you don't need the quadratic formula even this technique can be used to solve any equation although you might be saying these numbers here it seems like we always had the signs the pluses and minus signs in the right place well let's try something else what if I switch up the signs a little x squared plus 6x minus 4 equals to 0 just to review what we're looking for we want to factorize this as x squared plus 6x minus 4 equals X minus something times X minus something we're hoping that we can find this kind of factorization at this point I should also say it's kind of miraculous we always seem to be able to find this factorization and that's because we were able to take square roots of negative numbers that's why complex numbers were useful and you might wonder why do we want to try to factor into multiplying two of these well if you tried to factor with multiplying three things like this you'd actually have an X cubed term it just wouldn't work okay so now in order for this factorization to exist what do I need I'll write my whole thing again if can find two numbers with some equals this time the sum is going to be minus six and that's because I'm insisting on putting these minus signs here putting minus signs here is what lets me eventually say that the numbers in these blanks are the exact solutions to this quadratic equation to begin with because if you remember we use the fact that multiplying two things to get zero happens precisely when at least one of them is 0 and X minus blank is 0 precisely when x equals blank so I need two numbers with some equals negative 6 product this time the product has to be minus 4 minus this blank times minus that blank is just the product of those two blanks and that's supposed to be minus 4 then they are all the solutions okay so I just need you so that half of minus six which is -3 minus U and minus three plus u give the product that I want if I try multiplying these again the beautiful thing happens actually this beautiful thing is called a difference of squares because that's exactly what we're doing the difference of squares is 9 minus u squared being negative 4 now how could I get that to happen I can make that happen when we do squared equals 9 minus negative 4 which is 13 and so I could make that happen by choosing you to be square root of 13 again choosing minus square root of 13 is also okay if I put it up here then I find that my two solutions are minus 3 plus or minus the square root of 13 and it works again so even if these signs are not the exact beautiful pattern it's okay just think about how this factorization works and just follow your nose and you'll figure out what the sum and product would need to be in order for the factorization to exist and again every time since at the bottom I get a u that exists that works I'm able to go all the way to the top get my if to be true and then then they are all the solutions ok so far we've still been lucky though we're always dividing this second number by two we're always taking the sum and turning it into an average what if it's odd let's try another one x squared minus X minus 1 equals 0 actually the same thing works if can find two numbers with some equals to one and product equals to negative one then they are all the solutions okay how would I get some equals to 1 then I could do that by having average equals to 1/2 so I need to find a u so that 1/2 minus u and 1/2 plus u give the product that is precisely equivalent to when I take the difference of squares 1/4 minus u squared being equal to negative 1 how does it get that I could get that by having u squared equals 1/4 minus negative 1 is 1/4 plus 1 that's 5/4 and so I could do this by having a u which is the square root of 5/4 I'm going to write it like this root 5 over root 4 and that's because root 4 is 2 so that means that my solutions here are 1/2 plus or minus root 5 divided by 2 actually oftentimes people will collect it over a common denominator and write 1 plus or minus the root 5 all over 2 and again since I got you to exist all of this logic flows I'm able to get two numbers with a sum of 1 and the product of minus 1 BAM and those are the solutions actually this is a fun equation if you if you like math one of these solutions is what's called the golden ratio which is the ratio of the length to the width of the most beautiful rectangle in the world let's do one more complicated equation just to see what happens if the front is not just x squared what if I have something like 2x squared minus 4x minus 5 equals zero Oh No then everything breaks right well not really actually the beautiful thing about algebra is that you can do anything you want to both sides of an equation to try to make it into something you'd rather deal with and we'd rather deal with stuff which has just an x squared without anything in front of it one x squared so just divide everything by two this equation here is equivalent to equivalent means it has exactly the same solutions to because I just scaled everything by two x squared minus two X minus 5 over 2 equals zero every solution of this is a solution of that and vice versa because to go from this equation to that one I could multiply both sides by two and to go from this one back down I could divide both sides by two both are legitimate things to do to both sides of an equation okay so now if I have this we just move back into the framework we've had before if can find two numbers with some equals to two and product equals two minus five over two then they are all the solutions following what we've done before we just need you so that one - you and one plus you give the product and that's the same thing as 1 minus u squared being negative five-halves okay how to get that just as before I could get that by having au squared which is 1 minus negative 5 halves which is 7 halves and we're just using fractions actually there's nothing special about this it just means that we have to deal with fractions instead this happens when I choose au which is the square root of 7 over 2 I'm going to write like this root 7 over root 2 so the solutions to this are just 1 plus or minus square root 7 over square root 2 I will say that sometimes people prefer to write this in a little bit more pretty way where you don't have a square root in the denominator that's called rationalizing the denominator and if you multiply the top and the bottom of this fraction by root 2 that's the same as 1 plus or minus root 14 over 2 both of these are actually the correct answer it just depends on whether you are supposed to give your answer in a form that has no square roots in the denominator at this point we've seen that actually this technique can solve every quadratic equation because this technique will always come down to just take a square root at the end and that actually lets you derive the quadratic formula if you want let's do it this way let's start by deriving the formula for something like x squared plus BX plus C equals 0 if you can find two numbers with some equals minus B and product equals C that's actually what we have to do here then they are all the solutions so just as before in order to get a sum of minus B I just need to have an average of minus B over two to get that so I need a u so that minus B over two minus U and minus B over two plus you give the product which is exactly the same as B squared over four minus u squared equals C how can I make this happen I could make this happen by having u squared equal to B squared over four minus C and the way I could make that happen is by having u equal to the square root of B squared over four minus C again if I chose negative square root that's also okay that give me a solution set to but it would be the same solution set and the very important thing is that this actually always exists because no matter what B and C are I can always take the square root of any number I might just get an imaginary number putting it all the way back up here I find out my solutions to the quadratic R minus B over two plus or minus the square root of B squared over four minus C that might not look like the quadratic formula that you're used to it's actually equivalent to it but if I really wanted to get the quadratic formula that you're used to learning we want to do that with something in front of the x squared as well so for the grand finale let's now do it with ax squared plus B plus C and we're actually going to use this formula here as a shortcut if I have a x squared plus BX plus C equals to 0 just as before all we have to do is divide by the a to get an equivalent equation here I'm confident saying divided by the a because if a was 0 then it wouldn't be a quadratic equation anyway and you have an easier way to solve it so I'll have x squared plus B divided by a X plus C divided by a equals 0 and these two have the same solutions for X because from this equation I can times a to get that one and for that equation I could divide by a to get this one because I'm saying a is not 0 in everything I'm doing now to get the solutions of this you could actually do the same thing we are done with this summon product but the shortcut is to say this is already in the same form as what we have found formula for you see it already is x squared plus something X plus C so I can just use the previous form where I actually will put that B is equal to B over a little bit over a and the capital C in the other form is the little seed over a I know I have big and little letters so I'm trying to emphasize that's a capital C and that's a lowercase C but if I use these two that is x squared plus capital B X plus capital C equals to 0 and so from what I did before I know that the solutions are it was negative capital B over 2 which is negative B over 2a plus or minus the square root of capital B squared over four if I do that I will get little B squared over 4a squared and I needed to minus the capital C so that's minus the C over a this is the quadratic formula right again it might not look like what you're used to to be honest if you plugged in A's B's and C's you would get the same answers but let's clean this up and you'll get something that you're used to seeing the easiest way to clean this up is to write that the beginning is still minus B over 2a but the next term what we'll do is we'll put everything over a common denominator 4a squared that leaves B squared here and when I say a common denominator if I want this to be over 4a squared then there will be a 4ac over 4a squared so that's for a see okay if I want to continue simplifying I can use the fact that the denominator 4a squared is exactly the square of 2a so I could also write that this is minus B all over 2a plus or minus the square root of b squared minus 4ac all over 2a there is a little technicality here and that I said I could write instead of the square root of 4a squared right just plain 2a actually you don't know whether it's the plus or the minus square root but the thing is it doesn't matter because in front I'll have a plus or minus sign anyway finally I can make this look a lot more elegant by combining everything over a common denominator of 2a and so that is just negative B plus or minus the square root of b squared minus 4ac all over 2a that is the quadratic formula that everybody learns although we got to the quadratic formula to end up this video I want to emphasize that the goal of this video wasn't just to derive the Radek formula so that we can now memorize it rather it's better to see that some parts that used to be done by guesswork have been replaced by a clever trick and also you can now use that to reason through every single step of solving a quadratic equation even if you forgot the quadratic formula here's all you do you start with an equation if it doesn't look like this just divide by what you need to make it have nothing in front of the x squared meaning it's just 1x squared and then the key is because of factoring if you can find two numbers who have a sum and a product that correspond to these coefficients these numbers in front of the X and this constant this thing with no X's if you can find those two numbers then those two are all the solutions I want to actually at this point say these ideas aren't actually new this particular relationship between the sum and the product of the two solutions of an equation and coefficients is actually often called v.a.t.s relations and that was known hundreds of years ago v8a was a Renaissance era mathematician so in fact what we've just done here is we have also proved vieta's relations for quadratics for things which are with x squared as the highest power of X so there is a relationship here between vieta's relations and this method this method establishes the ATA's relations but I should say this technique has been known for hundreds of years this trick I also want to emphasize this particular trick is also not new the trick of saying if I have to find two numbers with a given summon a product that was actually done thousands of years ago by the Babylonians the Babylonians knew that if you wanted to solve these problems the easiest way would be to go from the average plus or minus a certain amount so this particular technique is actually just the combination of two tricks or insights that have been known for hundreds of years and thousands of years that's what makes it all the more surprising that this method hasn't been the one that we all learned in school hopefully by combining these insights from the ancient mathematicians all of our future generations will not have to do any more guessing [Music] [Music]

Channel: Daily Challenge with Po-Shen Loh

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Rating: 4.9091754 out of 5

Keywords: quadratic, math, quadratic formula, quadratics, solve quadratic, factor quadratic, factoring method, trick, factor, po-shen loh, loh, new, easier, different, simple, simpler, learn, how to, learn how, explain, example, examples, method, expii, daily challenge, mathcounts, math contests, math competitions, Quadratic equations, Easiest Way to solve quadratic equations, Solve quadratic equations by factoring, Quadratic equation, How to solve quadratic equations, Solve quadratic equations, when b=0

Id: XKBX0r3J-9Y

Length: 40min 5sec (2405 seconds)

Published: Wed Dec 11 2019