A Level Maths Free Tuition Lesson - Completing The Square and Discriminant

okay welcome back everyone to yet another uh free a level um tuition lesson for a level maths that I'm doing Live on YouTube so these are done every single week Thursday 6 p.m to 7 p.m and today we're going to be doing completing the square and discriminant now completing the square you may have seen back in um year 10 and 11 so your gcses but it has slightly changed okay so it can be slightly more complicated so first of all with completing the square what is it well it's just another way that we can write quadratic so for example if we have x squared plus 4X um minus 3 equals zero we can write that in a few different ways so we can use the quadratic formula we can factorize it into brackets we can do all of this fun stuff completing the square is just another way of doing it so you may Wonder well why do I care well the reason why you should care is number one there are exam questions which literally just say using completing the square show that or make it this the second point is you can actually find what the minimum of any quadratic is which is actually pretty useful okay so when you're sketching them you can work out the minimum Point without having to do some uh interesting nonsense okay so let's work out how to do it well as you know with quadratics we like to label the number in front of the x squared as a the number in front of the X says b and the number at the end as C so for the completing the square we have a little formula and it's X plus b over 2 squared minus B over 2 squared and that's it okay you can say plus C if you like but technically the plus C isn't part of the formula okay so if we were to apply it to this case we would have X plus 4 divided by 2 so 2 squared minus 2 squared minus 3 so then expand this out we have X plus two I should say equals zero X plus 2 squared minus 4. if you look this is always going to be negative because you square it to make it positive then make it negative minus 3 equals zero so then you have X plus two squared minus seven equals zero and I forgot to write the two cool so we've done completing the square you just apply this formula perfect so then what else do we need to do or what else can we do well we can also find what the minimum point is and the minimum point is just going to be so the minimum of this one all you do is you flip the sign of this and make it minus two and you keep the other sign the same that's it that's how you can work out minimum points okay that may not seem too difficult but you know it is a thing that we can do it's somewhat interesting and that's pretty much all there is to completing the square this discriminant thing will come up onto a bit later but for now let's do some practice with completing the square to see what we can do so show that this thing is true for all real values of X so unless you do further maths you don't need to care about the for all real values so we have x squared minus 6x greater than zero right so what we can do is actually once again use our bracket so if you remember we half b so be minus three squared minus minus 3 squared plus 11 is greater than zero so now we can start simplifying minus 3 squared minus 3 squared is nine make it a negative is minus nine plus eleven zero okay now in terms of the reasoning there's actually a few things that you can do so the good thing about being in sixth form now or doing a levels anyways you get a bit more freedom so one way you can prove so if we say way number one is you can say that the minimum is and again flip the sign and keep the sign the same too this is greater than zero so the the lowest point on the graph so if I were to draw this quadratic out the lowest point is at three comma two and I recognize that that's not in the right quadrant okay it's just for demonstration which means it never goes into the negative which means it always has to be positive which means it's always greater than zero one other thing is what's the smallest thing this can be well the smallest thing I can make here is zero right because if I make this a negative number it gets squared and become positive if I make it a positive number it just becomes a bigger positive so the smallest can be zero then I'm adding two of course that's greater than zero both of those work okay so in that case minimum is three over two which is greater than zero is the way I'm going to say it's the way I'm going to describe it as being um greater than zero cool okay so I want to hide that what about this well here we have prove that for all values of x squared plus 4X plus 12 so let's write this out oops okay not sure what that was about minus 4X plus 12. so let me Zoom back in right plus 12 is greater than X Plus 3. so the question is here how do we do this well it would be really easy if it looks something like that so why don't we make it look like that so if we get rid of everything on this side we will then have something that looks like the previous question so in minus X and minus 3 from both sides x squared stays the same minus 4X minus X gives you minus 5x 12 minus 3 gives me 9 is greater than zero and then we do the exact same thing we just did so we have X minus five over two squared minus minus five over two squared plus nine is greater than zero you can use a calculator to work this out if you want to because it's a calculator paper but whoops so why don't we choose a calculator you don't need to you can work it out using fractions if you prefer we have minus minus five over two squared and then plus 9 on the end which gives me eleven over four I'm going to leave it as a fraction because it looks a bit better so again if you want you can say the minimum flip the sign of what's in here you get five over two eleven over four which once again is greater than zero or you can see that the smallest bracket can be a zero and we're adding a positive number which means it must be positive overall okay so let's leave that as it is for now okay now let's do this next bit find in terms of the constant P the roots of the equation this is pretty interesting so with completing the square one other thing we can do is calculate answers right we can actually calculate what the roots are going to be okay and this question is actually saying what the root is going to be so if I go back up to the top for a second I have this solution here right I can find the roots to this equation now the reason why this is useful is sometimes we don't have easy numbers to work with when we factorize so if I plus 7 to both sides you're going to notice we get assert and this isn't unique for completing the square generally speaking we get thirds square root both sides we get X plus 2 equals plus or minus the square root of seven then we subtract 2 from both sides we get x equals minus two plus or minus the square root of seven now that looks like what you'd get from the quadratic formula right which is the point you can do this and get the same answers you'd get with the quadratic formula but I haven't used the quadratic formula and in my opinion it's a bit easier with this question they're basically asking us to find the solution so we can either use the quadratic formula or completing the square but in this case they tell us use completing the square so that's exactly what we're going to do so by completing the square so again we just load up load up we just think about how we do completing the square we have X plus half of B which is p over 2 squared minus p over 2 squared plus 4 equals zero now let's go ahead X Plus p over 2 squared minus p squared over 4 plus 4. and this is where it's going to get a bit spicy move this over to The Other Side by doing the by flipping the signs whoops over four minus four okay and now we're going to square root both sides and this is where again it's going to look ugly but this is what they've told us to do X Plus p over 2 equals plus minus remember p squared over 4 minus four and what was the last step I did I just subtracted this from both sides in the previous bit and I normally put that first because it looks better okay just because you're doing maths doesn't mean everything has to be ugly you've got it oh okay now this next part kinda ties in to the discriminant okay kinda so this has no real roots when this thing here in this square root is less than zero I want you to think for a second think about why that is the case okay real Roots is the point here okay so for me to have no Solutions X can't equal anything I have to get a math error so the only way to get a math error is if I try and square root a negative number I.E this thing here is less than zero okay let's try it right now right like I can square root basically any number but it breaks down if I try and do the square of minus one so that's the only way I can get math error is if this is negative if it's zero like a zero it's positive like any number whatever it still works the only time it doesn't work is if it's a negative so real Roots when it says no real Roots it means there's no Solutions which means this has to be negative so now let's do it let's add 4 to both sides we get p squared over four is less than four times both sides by four I get P Squared is less than 16. and then go over here square root both sides I get p is less than plus minus four and that's it that's your answers so P either has to be greater than minus four or less than plus four that's it not bad not bad now the way I did this was a bit interesting we'll go through that in a bit okay when we do sketching but you can tell which side the inequality is going to take by what the graph would look like which okay next question oh God my voice went okay so the curve C has the equation x squared plus ax plus b and it says given that the minimum point is four minus three find the values of A and B so where did we hear about minimum point well the minimum point is oops I've just scored another thing the minimum point was just when I swapped around these things here so if we go up here we flip the sign of this and we took this to be the real thing so let's do that I'm going to just write what this completing the square must look like x minus 4 because there's a 4 here squared and then -3 equals x squared plus ax plus b again I'm going to go to that one more time the reason for that is to work out the minimum we flip the sign of what's inside the bracket and we kept the sign of what's outside the bracket so all I've done is using this information is I've worked back I reverse engineered my solution okay and I say that because it sounds cool so what we're going to do now is expand the left side and make it look like the right side so x minus 4. x minus 4. minus three let's expand this I'm not going to write the right hand side for now x squared minus 4X minus 4X gives me minus 8x minus four times minus four is plus 16 minus 3. x squared minus 8x plus 13 and we know that that still equals the stuff from the top so now you just need to compare a is the number in front of X the number in front of x is minus eight B is the number on the end which is 13. those are my Solutions not too shabby I hope you find okay okay now this this is the one so that looks a bit weird right now whenever you see something like this right in the form blah blah it's always completing the square because it's the only way we can get something that looks like that but this a and this 2 should spark some interest okay because when you have a number in front of x squared you're going to have a number outside the bracket okay so if you remember if we go all the way back up there was no number in front of x squared when we do completing the square there has to be no number in front of x squared so or you can say the coefficient of x squared has to equal one so now how can I make the coefficient equal one well if there's a 2 in front of it I can divide by two now because this doesn't equal anything I can't really divide it by two that easily but I can factorize 2 out and then what I do is I do completing the square that's how I do the answer so two and then if what I like to do is when I have a bracket and then a bracket next to it I like to use stable brackets for the outer bracket and the inner one gets a regular curved brackets because I think it makes it look a bit more clear so X plus two squared minus 2 squared and then plus a half okay by the way there are other ways to do it um and what I mean by that is some people like to factorize just the beginning bit and leave the one outside and add it at the end totally fine it's the same thing either way you want to do it so two X plus two Square minus 2 squared so 2 squared is 4 sticker minus again I'm doing this in like really slow steps and I recommend everyone does this for now because again we're not in the exam yet we can take our time and be a bit sure that'll be minus 3.5 I'm not going to write it as a fraction but it's 7 over 2 if you want it as a fraction um which can help now we expand the brackets because look here there's no square brackets so we get two X plus 2 squared minus seven that's it so if you want to find the values a is to B is to C is minus seven not too shabby so if we keep going I believe okay yeah we've got a few more questions before we get on to the um discriminant discriminant really isn't that bad I think you'd be fine with it but is away let's do it again this means comp Square because it's the only way we can get a quadratic in this form so let's do it we have X Plus 9 over 2 squared minus 9 over 2 squared plus three okay so let's do this again you can use a calculator so don't really worry too much about it but 9 squared is 81 2 squared to four so you can just leave it as that and let's get down to business so we have minus um nine over two is just 4.5 so let's write that minus 69 over 4. done two marks easy peasy lemon squeezy State the coordinates of the minimum point of this curve so I just went through puberty of this curve well it's the exact same equation as the question so Y is just equal to X plus nine over 2 squared minus 69 over 4. so the minimum point if you remember you flip the sign of this you get minus nine over two and you just keep the same thing so minus 69 over 4. that's your answer four marks in about two minutes if that not too shabby at all not too shabby at all okay maybe one second I was gonna eat some water before we do the next question okay okay so by completing the square find the constant K of the roots of the equation well let's do it so once again this is similar to the one we had before we're going to get something that's just plain hideous right so completing the square gives us X Plus K over 2 squared minus K over 2 squared minus 6 equals zero simplify this out minus K squared 4. 4 minus six zero now to find the roots Roots just means Solutions why do they have two different names it's a very good question moving on move all of this to the other side except the brackets so X Plus K over 2 squared again you probably remember this from GCSE as it's good to know that there is some like somewhat easy content in a level it's good to go over it now we square root both sides and we get X Plus K over 2 equals now the best way to do this let's leave it just as is yes you can square so basically keep in mind that you can't just square root these two things separately okay so you can't just do Square K squared over four plus square root six it's not the same thing so just be careful of that then minus the K over 2 we get x equals minus K over 2 plus minus the square root of K squared over four plus six and that's your answer four marks not bad I personally think but too bad and then it says hence find the exact root of the equation so hang on a minute okay this is kind of annoying this question is missing something it tells you what K is and in this case it should be four so I recognize the question it should have cables 4 somewhere but it doesn't I think that's my bad yeah okay over four so you just need to find the solutions again this isn't bad at all subbing K so minus four over two plus minus six I don't know why I wrote it like that so that's going to give me minus two plus minus uh four squared over four just give me four plus six is ten so our two solutions are x equals minus two plus the square root of ten or x equals minus 2 minus the square root of 10. that's it not too bad ah okay cool so what we need to do now is work out or not work out sorry we need to do the discriminant okay so what is the discriminant that's a very good question the discriminant tells us the number of unique Solutions to a quadratic okay so I'm going to write out the quadratic formula cool this formula is extremely powerful it tells us all of the unique solutions for a quadratic whether it be no Solutions one unique solution or two unique solutions by unique I mean that sometimes you get you know when you sub this in and you get x equals like four or x equals four it's not a unique solution there's always two solutions but sometimes you know or it's unsolvable but sometimes they're not unique there's a part of this equation that determines whether or not we have zero one or two solutions and that is this a bear with me if the thing inside so B squared minus 4AC if it's negative what happens let's try it out I'm just going to sub in random numbers right so let's do minus 10 plus the square root pick a negative number let's do negative five over two oh no math error because we can't square root a negative there are no real and unique Solutions okay does that make sense hopefully that makes sense now what if it equals zero well if this equals zero the whole equation collapses into just being x equals minus B over 2A that only has one answer so there's one real and unique solution because this only has one answer the plus minus is what gives us two answers right now what if it's positive well you can probably guess that there are two real unique Solutions and the reason for that is when we square root a number we get two values so when I square root 4 right so let's say it's positive the answer is 4 I can have plus 4 or minus four so I have two different answers so I'm either going to add four and then divide or take away four then divide which has to give me two different answers unless it's zero which is what we have here but if it's positive it has to give you two answers two different answers that's where the discriminant comes into play and you can be asked to do various things okay so one common thing is it has no real Roots so find the set of possible values well if it has no real Roots then that means B squared minus 4AC is less than zero so no real roots means B squared minus 4AC less than zero so let's sub that in we have K squared for B minus four a is one C is two is less than zero so K squared minus four times one is four times two is eight is less than zero so K squared is less than eight so K is um okay it's not it's less than plus minus eight oh God that was a terrible mistake please forgive me that's why it's the square root of eight okay now the problem is here we have two solutions okay now if I draw this out like draw this graph out okay it's positive so it's going to be a u shape yeah like this this here would be the minus root eight this here will be the Plus root eight if I want less than I'm looking at the bits between it so we have minus root eight is less than K not equal to less than eight does it make sense that it's less than or not equal to yes because if it's equal to then it would be on the line which is zero we want negative so Less Than Zero that's it question one not too shabby at all here's where things get a bit more interesting equation KX square plus 5x plus k equals zero where K is a positive constant this is now important information so I've given a bit of a spoiler to the answer to this question has equal Roots equal means that there's one unique solution it's just another way they like to write it equal Roots because both roots are equal to each other so there's one unique solution which means B squared minus 4AC equals zero so once again B squared we have 5 squared minus four times K times k equals zero so we have 25 minus 4K squared equals zero let's move the K squared to the other side because I don't like negatives uh for my variables then we can divide by four and then we're going to square root so square root of this becomes plus minus square of 25 is 5 square root of 4 is 2. okay find the value singular positive constant so k equals 5 over 2. students drop a mark all the time by stopping here the questions asking for one value because there's one answer because it's a positive constant students every single year all the time always decide to um not read the question properly and give two answers it has to be one answer or you drop a mark this is a three mark question so you lose a third of the marks which is um bad okay so please keep that in mind so as for the next question let's keep moving onwards to see how much okay good no real roots right so no real Roots no Solutions means that b squared minus 4AC is less than zero because we're Square rooting a negative which gives us math error which means there's no answers find this set of possible values so it's again it's a set so B squared we have 6K squared make sure you square the whole thing minus four a is K C is two zero 6 squared is 36 K squared is K squared minus 8 k equals zero now what I can do here it's not quadratic well it is but I can get rid of the quadratic by dividing by 8 k so the biggest thing I can get through both of these is um 8K okay um sorry divide by K right so I can cancel through by K we get 36k minus 8 equals zero so 36k equals eight k equals 8 over 36 which again you can use the calculator or your brain to simplify down top and bottom by four you get two over nine now it says a set so actually I should have wrote Less Than Zero that's my bad so K is less than two over nine uh the reason why I made that mistake just now writing equals is I some I usually teach pupils that don't like inequalities to just write equals and then add the end at the inequality back okay which is fine if you want to do that it doesn't actually change anything but that is the correct way of doing that perfect okay so hopefully that's all okay again just stop me the second you're not sure you want me to go over it again um we went through completing the square a bit earlier for those of you that have just joined um and that there's some interesting questions and we've done the discriminant I'm going to do a summary at the end of the lesson where I scroll through the questions so you can ask any more questions so be sure to stick around for that if you missed the first part of the lesson okay this is where it probably gets as hard as it gets in terms of the um discriminant okay two distinct real Roots so I'm just gonna I'm just looking at that now B squared minus four AC greater than zero that's what I want to start with okay now the difficult bit is the fact that A and C are expressions which look a bit tricky but it's actually not that bad okay so what we're going to do is substitute it in B is 4. minus 4 a is K plus 5. C is K plus 2. greater than zero so we're definitely going to get quadratic so let's get down to business 4 squared is 16. now I'm going to leave minus 4 on the outside K squared plus 5K plus 2K so 7K plus 10 great zero now we can times it all by minus four some students do this all in one step and they mistake and they get the signs wrong okay they mistake the signs seven times four is twenty-eight minus 40. now we can actually simplify this make this minus 4K squared minus 28k 16 minus 4 would be minus 24 it's greater than zero so what we can do here is we can do something quite special all of this is negative so what I can do is just Times by negative one or move everything to the other side or divide by negative one but the point is I'm making all of this positive okay positive numbers are easier to work with generally speaking next thing we're going to do is I actually notice I can cancel this down by dividing everything by four K squared 28 divided by 4 is 7. 24 divided by 4 is 6. now I can factorize this two numbers at times 2 6 and add to seven is six and one so K plus six K plus 1 is greater than zero so it crosses the x axis at minus six and minus one and now I need to know what side the inequality takes okay so the way to do that is you draw your inequality uh your graph out and it's going to cross up minus six and minus 1. and it's a positive quadratic so it looks like a big happy face okay because it's a positive quadratic and I'm looking for the values that are above the x-axis so K is either less than -6 and K is more than minus one because I'm looking for the things or the values that are outside of this range okay but outside of that range I hope that makes sense okay so this question is basically the same as the previous question so let's go through and see if you guys retained what you're what we're talking about two distinct real Roots B squared minus four you see zero so we have n plus 1 squared minus 4 a is one C is three minus three n is greater than zero n plus 1 squared we could expand that to N squared plus 2N plus one and minus 4 times 1 does nothing so 3 minus three n is greater than zero so we have N squared plus two n plus one minus sorry 12 minus 12 n is greater than zero so we have N squared minus 10 n minus 11 is greater than zero so you can't actually do this without using the quadratic formula okay so you have to use the or you can use completing the square so quadratic formula we're going to use x n equals minus and actually let me write out the quadratic formula for you just to as a reminder plus or minus the square root of B squared so minus 10 squared minus 4 a C there we are now you could use your calculator so that's one way to do it or you can actually simplify this and get the third um again it really is up to you so so I can actually type this minus wait can I oh my goodness okay I'm learning more about this calculator every day plus square root I don't know how to do them by the keyboard mine's ten squared so I get just make sure that's all so yeah minus 10 n okay wonderful so I get 11. or n equals also one thing that really triggers me just just like use the arrow keys to go and change the thing oh no I said that and now I've just messed up no oh God all right basically use the arrow keys to change what the value was and don't be stupid like I just was so I right and change it to a minus this time just change that you don't need to change the whole thing and rewrite it you don't want to be like me in this case don't rewrite it it was a bad move oh minus one oh that was disappointing okay cool and again we're looking for the values that are above the x-axis so this is our quadratic we have 11 and minus one and a big U shape and we're looking for the values above zero so that's this side and this side that happens when n is less than -1 and N is greater than 11. so that's our answer so uh later as for your question um I stream every Thursdays between six and seven PM for a level maths and for GCSE maths is every Tuesday six and seven pm perfect perfect okay again this looks like a very familiar question right so it should because of doing the exact same thing but it has no real Roots so B squared minus 4AC is now Less Than Zero so what we're going to do here is we're going to write this out so 2K minus 3 squared minus 4 1. and K plus 3 is whoops Less Than Zero okay uh just make sure I've done that right because I've got distracted okay perfect so we're going to minus 4 again K plus 3 because 4 times 1 doesn't do anything zero expand it 4K squared minus 6K minus 6K Plus 9. minus 4K minus 12. is less than zero okay hopefully that should answer your question Okay so next we have to bring this all together so 4K squared there's no other K squared let's leave it as 4K squared minus 6K minus 6K there should be okay here too is minus 12K minus 4K is minus 16 k 9 minus 12. is that right yeah yeah good nine minus 12 is minus three that's not zero okay so just make sure I've done this all right so with nine minus three that's all good yeah all good so all we need to do here is once again we're actually going to use the quadratic formula okay um and it's going to be pretty damn ugly unfortunately but that's just the way it is K is equal to minus B so minus minus 16 plus minus square root minus 16 squared minus 4 a is four and C is minus 12. over 2 times 4. again let's just uh enter the whole thing on our calculator and this time I'll try and not be stupid and remove it ah perfect okay so we can actually change this like this I like to do the plus first I'm actually not oh thank God all right I took a risk there and it was unnecessary but I did it anyway minus minus 16. change that to A6 good first answer is 2 plus root seven you can probably guess what the next answer is going to be 2 minus root seven cool now we need to work out which way is um going to be now you might be noticing a pattern some of you that are kind of Fairly on it but if you're not then just keep drawing it out two minus root sevens over here two plus three seven will be equally distant away but actually it wouldn't it would be slightly further away and we have a u-shape perfect we're looking for the numbers under the graph because it's less than zero so that happens between the values of 2 minus root seven less than K less than 2 plus root seven okay not bad not bad at all okay so what we could do is we might not have time for a summary if we do some more questions but if you want a summary keep in mind that these videos a the live streams are saved on YouTube but I also edit a version and upload it the day after so on Friday so if you can't make it for the live stream be sure to check the channel on Fridays in the evenings and I will upload kind of an edited version and that does help I just realized I probably need to actually make the lighting a bit better in my room no no just to make sure it doesn't uh look all weird I look like a ghost at the moment uh anyways enough about my problems the curve with equation this does not intersect with the equation 2x minus 12. so when we normally talk about an intersection right we equate the two curves right now seeing it says it does not intersect this equation has no Solutions which means we can discriminant okay move everything over we get p x squared I get so what I want to do here is I want to make this actually no I'm not going to do that in one step because that's a bit stupid otherwise it will confuse you guys minus 5p equals zero now this has no Solutions now I'm going to do something just to make the discriminant slightly easier I'm going to factorize that a minus four minus one from this I make it 4p plus 2 x and then just minus 5p as well no Solutions means that b squared minus 4AC as you guys are an experts at is less than zero good so keep in mind this is our coefficient C sorry not coefficient RC B squared about one second my voice has just died okay I think I'm back in the land of the living that means we have minus 4p Plus 2. all squared because I'm squaring the minus as well help me okay minus four a is p C is minus five P plus 12 should be less than zero yes it's pretty ugly right so again it's ugly but that doesn't mean it's wrong so to speak okay so if I'm squaring this I can just make that positive right and then expand it out so make that positive and then just expand it out minus 4p minus 5p plus 12. 4B times 4p is going to give me 16 P Squared plus 8p plus 8p gives me plus 18p Plus 4. minus 4 times minus 5 is plus 20 P Squared minus 4p times 12 is going to give you minus 48. p um oh that falls really annoying I could have divided it all by four okay 16p squared plus p squared gives me 36 P Squared minus 18 plus 48 minus 30p 4. and then depending on how you want to solve it you can use the quadratic formula again jeez okay minus minus the 30. plus minus six again apologies if I'm getting a bit quiet my voice is being punished routinely I've got those numbers right around looks like I have well no hope but pray and once again I deleted the thing because I'm a an idiot start with plus because I like to whoops God oh okay that's pretty reasonable uh two-thirds and hey look I didn't make the same mistake again I'm 1 6. so I'm hoping maybe some of you by now have noticed that when it's less than zero and this number's positive it's always the inequality is always between those two values but for those of you that might not be convinced we have a sixth two-thirds we're looking for under zero so that's below the x-axis so these values here so again it's going to be a sixth is less than p is less than two thirds okay jeez um one other time we used completing the square is with equations of circles and we're actually going to cover that in an upcoming lesson so what I'll do is I'll leave that for now because we don't really have time to just go through it and like slam another thing on you foreign so um I thank you all for coming out coming out I thank you all for tuning in uh again just be sure that you're kind of here on time again it's going to be 6 p.m 7 P.M every single Thursday and that's going to be UK time uh the videos are uploaded the day after and no worries Lila hope you found it useful um but again we're coming on to the kind of uh Circle theorems like this equations of circles a bit later but for now this will do again so I hope you all have a really wonderful evening and bye