A Level Maths Made Easy - Completing the Square

hello and welcome back to another one of these preparing for a level maths video so once again if you look in the link in the description there is going to be this worksheet that you can access and you can download again I'd recommend going through this video now and then doing the worksheet a day or two afterwards just to see how much you've retained but we're going over something that again we've covered before in GCSE called completing the square however there are a few kind of new pieces as per usual as always that will come up in as now I want to preface this technically this isn't everything that's in completing the square and the reason why I left this out I did the beta I had it in originally and then I deleted it because you also use this with equations of circles so what they'll do is they'll give you some long expression and they'll ask you to work out the equation of the circle and you have to use completing the square for that now the reason why I left it out is because there's going to be a video on equations of circles so I'll just do it then as opposed to going over this now but this again illustrates my point that it's very hard to look at subjects or topics individually because they all link together and come back again and again and again you will be expected to recognize when you can use completing the square and when you should now completing the square is basically another form of factorizing okay so you can use it to plot graphs you can use it to find the minimum or maximum points and you can use it in order to solve them if you'd like as well so if you want to give an answer in third form less common because of course you don't have a non-calculator paper so realistically you wouldn't ever really need to complete the square to do it but if they ask you to you're expected to know it okay but just to keep that in mind that if you're wondering about the equations of circles and things like that that's in the equations of circles video that hasn't come out yet but in the future it will be of course so completing the square let's go with what that go what that means Computing the square is quite simple all you have to do is look at an algebraic expression like so so x squared plus BX plus C and you can write that as X plus b over 2 squared minus B over 2 squared plus C now if you notice and this is something that I really want to hammer home because a lot of students in their year 12 mocks get this wrong there is nothing in front of this x squared which means it's a one okay it's not minus x squared it's not 2x squared it's not half x squared it is nothing it is just the or the number one okay if you have something in front of x squared you factorize it out that's the key point that I really really really want to hammer on okay because a lot of students get that wrong so if we have a look at this it says show that this expression is greater than zero for all real values of X now I can already hear some of you typing you cannot use the discriminant for this the discriminant will just show you so if you those of you that haven't done the discriminant that's coming up in a later video as well but you can't just solve this equation right so if I solve this equation I'm going to get no Solutions right because it's it's greater than zero that doesn't show that it's greater than zero for all real values it just shows that it doesn't equal zero which is not the same thing okay because again we have graphs like so this is the one over X graph where it goes from positive to negative without going through the origin which means it never equals zero but it can be positive can be negative now of course you might say but it's a quadratic you're right but we're trying to teach you how that if you're trying to show that a function is always greater than zero you have to be very rigorous you can't just say it doesn't equal zero therefore it's not zero you have to actually show that it never ever goes down to zero okay it all goes below zero so we're going to be using Computing square and if we use this B is just negative six and C is 11. so just subbing this straight in we have X and of course it's a minus instead of a plus because it's minus six B over two what's six divided by two that is three squared minus again three squared plus 11. so we get x minus 3 now all we're going to do is expand this part over here one thing you can do and I would encourage you to do this is to just enter this bit into a calculator so you have minus 3 squared plus 11 right it gives you two minus three or minus of three squared is minus 9 plus 11 is plus two plus two now this is always greater than zero why because the smallest this can be so smallest is equal to zero now why is that the case well let's have a look if I make X positive then it gets bigger and bigger and bigger and bigger right so it's positive a positive number squared is always positive if it's zero I get 0 squared which is zero so it equals zero now what happens if I make this negative well a negative number squared is also positive so we have when X is less than zero it's positive when X is greater than zero it's positive or when the whole thing is uh kind of negative whole thing's positive it's zero above zero and when it's equal to zero to equal zero so the smallest this can be is zero and then we're adding two to it so zero plus two is always greater than zero of course right so if that makes sense when you do things in completing the square you get the minimum point so the smallest this bracket can ever be a zero right because if it turns out to be if x minus 3 is negative we Square it and make it positive if x minus 3 is positive we Square it and keep it positive but if x minus 3 is 0 then we square and it stays zero which means that is the smallest number it can be quite an easy one to start with this has prove that for all values of X this is greater than x plus three and I've seen students try and use completing the square on this side and then try and do some mental gymnastics it's not that bad if you ever have an inequality with letters and numbers on both sides move them all to the same side so I'm going to take all of this I'm going to move it to the left hand side so I'm going to minus X and I'm going to minus 3 from both hands both sides so x squared stays the same 4X minus X is 3x and 12 minus 3 is 9. so you always want it equaling zero on the right hand side and this is the same when you solve equations right you always want them equaling zero because that gives you the most flexibility just keep that in mind so again we're going to use this formula here so B over 2. B in this case is 3 and C is nine so we have X plus b over 2 so that's three over two squared minus and then that's going to be 3 over 2 squared again plus 9 and then it says greater than zero but we'll leave it as that for now so the next thing I want to do is I want to write this into my calculator so we have minus open bracket three over two squared plus 9 gives me 27 over 4 or 6.75 whichever you prefer so we have X plus 3 over 2 squared plus 27 over 4 and that is always going to be greater than zero because again the smallest this can be is zero which means again it's always been greater than zero right zero plus a number is always going to be bigger than zero all the time nice and easy I hope now here's where things get a bit interesting so I could tell like I have precognition a few of you probably found that boring but this is where we get some meat and veg so by completing the square find in terms of the constant P the roots of the equation oh that's a bit interesting but it's actually the same thing okay so this is where solving equations can be useful with completing the square again I said it's very unlikely they'll give you this because we have calculators however if they give you an unknown that's when things get interesting because how do you plug in an unknown into that equation the quadratic formula you can't so in this case we have B is equal to P which might be a bit odd and C is equal to four so once again we're just going to write down X plus p over 2 squared minus p over 2 squared plus 4 equals zero again it does say fine in terms of the constant P the roots of the equation so remember Roots just mean Solutions so what I want now I've got this is I want to rearrange for X that's the key thing that I'm thinking of so if I want to rearrange for X what I might want to do is instead of doing this square like squaring this I might just want to move this whole thing to the right hand side but again in case you didn't kind of think of that let's expand the brackets first so we have X Plus p over 2 squared minus p squared over four so again when you square bracket you square the numerator and the denominator so P Squared is P Squared 2 squared is four plus four equals zero now when we solve we want to move everything over to the right hand side so let's do that we have X Plus p over 2 squared equals p squared over four minus four now we're going to square root both sides but remember what you can't do is treat this individually so if I square root both sides right now which is what I'm going to do I can't write this wrong completely and utterly wrong okay instead you simply write a big square root and it looks a bit ugly I completely get that in a level maths we do get very ugly looking formulas and equations and answers but that's fine okay that's what you have to do then the lastly but certainly not least technically this would be plus minus by the way keep that in mind when we Square it now we need to put plus minus we have X is equal to we're minusing p over 2 to from both sides plus minus the square root of p squared over four minus four minus p over two and that's our three Mark answers no real way we can make that look better maybe we could put the minus p over 2 at the front so you might see answers where they instead do this and some people say well is it still a right if we have the first answer absolutely this is just purely a semantic change what I mean by that is I've written the same thing in a different way right it would be like me writing you know two times two versus two squared it's the same thing no one's going to grade you down for writing it in a different way okay unless they specifically say give it in terms of this cool now for Part B it says hence find the set of values of P for which the equation has no real Roots now this is where it gets a bit more interesting again so hence means we need to use part A so I'm immediately thinking we're going to use this immediately where it has no real Roots so we don't deal with non-real roots so just so you're aware there are always extra solutions to equations there is a solution to every problem essentially with quadratics however some of them are real some of them are imaginary now unless you're doing further maths you will not deal with imaginary numbers okay so don't worry about it but just through context so you understand where this kind of fits in imaginary numbers happen when you square root negative answers okay so if I square root negative one that is an imaginary number okay so when it says no real Roots effectively it's whenever you're Square rooting something it can't be less than zero for us regular maths people if you further maths you live in a different dimension okay so you have access to imaginary numbers and all that stuff but again for this context and by the way it's also an important thing further maths and pure maths if you're doing further maths you'll probably you I'm fairly certain you have to do regular maths as well at least that's the way I've always experienced it you can't use further matter in regular math right so you can't talk about imaginary numbers in regular maths so please avoid that so are there any limits for our Solutions absolutely there are whenever you have a square root there is a limit to your answers that limit is when this thing here inside here is less than zero so what I'm going to do is just find the set of values for which the equation has no real roots that happens when p squared over four minus 4 is less than zero so if we want to find the set of values and this is another key thing when it says set it means you're going to be using an inequality because it implies there's more than one value so if you get P equals two it's probably not the right answer you probably need some kind of inequality now if I solve for p all I need to do is add 4 to both sides so I get p squared over 4 is less than 4 times both sides by 4 P Squared is less than 16 square root both sides I have p is less than plus minus four so essentially we can't have P Squared being 4 right so P cannot equal plus or minus 4. actually that makes some amount of sense in that regard but effectively we can't have P Square level 4 being less than four here's another very typical uh year 12 mock exam question where you have completing the square and then immediately you need to apply your knowledge of completing the square so again if you'd like take a second to attempt this question for yourself again use this as active learning so treat it like a lesson I go through a question you attempt the next question check your answer and then attempt the next question check your answer attempt the next question check your answer okay unless it's something brand new it is why I thoroughly recommend if you actually want to learn this stuff so attempt it before I go through it so Express x squared plus 9x plus 3 in the form this nice and easy all we need to do is B is number is nine and C is three so we have X plus nine over two squared minus 9 over 2 squared plus three now what I want to do is just enter into my calculator this kind of numerical expression at the end so again we've got minus nine over two and again be careful with the kind of first of all if you ever want to square fractions they have to be in Brackets please for the love of God don't put the minus inside the brackets you get a completely different answer this thing here is always negative because when you square something it's always positive you put minus sign in front of it's always negative so be very astutely aware of that minus 69 over 4 perfect and now it says for Part B so that's two marks two steps Part B is so generous it is so generous State the turning point right or state the minimum point or the maximum point the reason why this is generous is because it's literally one step you should remember this from GCSE basically when you put something in completing the square you get the turning point you get the maximum or minimum Point straight away all you need to do and I'm going to show you the answer then I'm going to show you the reason why just like I did with GCSE the answer is this you flip the sign of whatever's inside the bracket so I flip the sign instead of plus it's going to be minus if it's minus it'll be plus and that's my x coordinate and then I write the exact same thing here whatever that is -169 over four that's two marks so generous so so so so so so generous so why is this well if you think back to the previous questions we did I mentioned about the minimum value of our like the smallest we can make our equation and what I did is I made this equal to zero okay so if I'm trying to make this bracket equal to zero I can write that as an equation I can say X I don't really need a bracket there let me get rid of that I'm going to say X Plus 9 over 2 equals zero so then all I need to do is minus 9 over 2 from both sides and I get x equal to minus nine over two and that makes that bracket as small as humanly possible which is why it's the x coordinate now if I have that x coordinate then this bracket is equal to zero what am I left with I'm left with whatever's on the end so that's why it works it's a really elegant simple way of finding minimum and maximum points again if you've done further maths you've probably done differentiation if they tell you in part A to use completing the square which is what they're doing here if you ever see this form it means completing the square 100 guaranteed completing the square so using that you should walk through it you shouldn't differentiate really although technically you'd be in the right once again we have these kind of weird algebra style questions so by completing the square fine in terms of the constant K the roots of the equation again that just means Solutions have a go please at doing A and B so I'll give you a few seconds let's go through it I'm assuming you've at least had an attempt let's go so first of all again by completing the square they're literally telling you what to do so just do completing the square B in this case is K C is -6 so we have X Plus K over 2 squared minus K over 2 squared minus six okay easy peasy line squeezy and we should make this equal to zero technically as well then we're going to again what we did last time it says find in terms of the constant K well that's what we did back up here and this again is why I love a level maths and I know that there are people that are going to disagree with me a lot of questions are the same thing they just change the numbers so if you practice this huge range of questions I'm trying to give you and then just try and do every question you can you're going to be prepared for like 80 of your exam and think about that eighty percent is a star a territory okay so you can always guarantee yourself an a if you do this so again what I did is I moved everything over to the right hand side so let's do that X Plus K over 2. squared and then I'd have K over 2 squared oops equals I'll expand that bracket now you could have expanded it first completely up to you really doesn't make a difference so we have K squared over four plus six square root both sides again we can't actually do the square root so we're going to just leave it in terms of a third and then again that'd be plus minus K squared over four plus six and then minus K over 2. uh I went one step too far that's actually the answer whoopsies then minus K over 2 from both sides we get X is equal to and again I'm going to show you the other way to write this so you can also have minus K over 2 first plus minus the square root of K squared over four again not the prettiest solution but it is the solution and that's four marks not bad right you get two markers for doing completing the square then three four marks like two more marks for rearranging not bad I'm hoping you're seeing right now I'm hoping this is making you not as scared um because the biggest jump is from GCC to I mean from year 11 cr12 was quite a decent sized jump that's not too bad hopefully and again if you put this minus K over 2 afterwards also valid don't let anyone tell you otherwise now for Part B I'm actually going to do this next to this equation this question just so we can kind of see it much more clearly it says hence so again immediately I'm just going to write out this exact same formula again because it says hence hence just means using the previous part in terms of maths anyway for those of you doing English you can probably tell me the actual definition of hence but for maths this is completely doable find the roots of the equation here now this is what I do the first thing I notice is that it's a different equation okay it's not K it's four and then I think okay it's different but how is it different well the X squared's the same the minus six is the same the only difference is that k equals four so what do you think I'm going to do I'm going to sub in k equals four that's it that's the intended solution and he says find the exact Roots so essentially all you need to do is do minus four over two plus minus square root of um 4 squared over four plus six so now what I would do is I'll just simplify this so we have minus 4 over 2 is just gonna be minus two plus minus ever unsure use your best friend the calculator in fact we can actually even let's actually see what it equals normally so four squared I'm just going to put what's inside the bracket for now it should just give me 10 so yeah it's gonna be square root 10. so plus minus square root 10. done right not too bad hopefully and now we're on the very last question of this one and this is again a harder style for two reasons the reason I love this question because a notice it's not one in front of x squared okay which means we have to use our formula should I have said that a bit later on when you've had a chance to pause the video and try yourself absolutely I should have said it later on but again hopefully you just paused it beforehand anyways um so we're gonna have to do a little bit of a trick there give it in this form cool if there is a letter in front of the bracket it means that there is a number in front of x squared just as a little bit of a hint okay and then we have some other business to do so let me show you how to properly deal with this so we have bear with me one second there's actually a few different ways to do it now I can leave this plus a I'm going to move it slightly further away and what I'm going to do is I'm going to factorize this first bit okay so I want there to just be x squared so I'm going to factorize anything that's not x squared so I'm going to make it minus 3 Open Bracket that'll make it x squared and what is 12 divided by minus 3 it is minus 4. okay and then we have plus eight on the end now we're going to do completing the square using this now here's something I want to preface some people like to also factorize out the eight so they like to factorize 3 from the 8 to make it minus eight over three completely fine so what I mean by that is you could also have something like this and you'll get the same answer do completing the square with this you'll get the same answer it's personal preference now if 8 was divisible by three so let's say it was plus nine instead I would do this I prefer it if it's not a fraction again completely up to you your choice don't worry too much about it and then we're going to do completing the square with this bit so we're going to have -3 I'm going to use staple brackets you're going to see me use staple brackets a hell of a lot during this a level maths course because when you have two layers of brackets it's very awkward to just put like another layer of bracket because you can't always see where one bracket starts and one ends and keep in mind we might have three layers of brackets in our answers especially when we come on to integration so I'm just going to put it out there the inner layer is going to be normal brackets then it's going to be staple brackets okay the third layer I use curly brackets but we don't really need that just now then we have another bracket we have x squared minus four so B is minus four we don't have a c that's completely fine so we have x minus 2 squared minus minus 2 squared and then the plus eight on the end let's expand the bracket on the right hand side so minus 2 squared is four that makes it minus again so minus four plus eight now we're going to expand the stable bracket so we're left with minus three x minus 2 squared minus plus even so minus three times minus four gives me plus 12 plus 8 so our final answer is minus three x minus two squared plus twenty again I write out every single step because it's very easy to make a mistake you guys have seen me make mistakes before I'm a human being believe it or not so it does happen on occasion but writing at every step individually helps your brain actually work through things properly lastly it has a turning point at M find the coordinates of M on you to everyone I don't care who you are stop the video only give you just take 10 seconds think about what you think the turning point is and what I'm going to do is I'm going to move this over in case you haven't paused it and that forces you to actually think about what you're talking about okay now let's go through it here's the answer just in case you guys want to check that's the answer this number at the front has no effect on the minimum or maximum point because remember we make the bracket equal to zero what is any number times zero zero so it does nothing doesn't absolutely bugger all okay likewise does this minus three affect the 20 no so it's the same method you flip the sign inside the brackets and that's your X and you keep this the same that's your y done and again if you wanted to practice again one key thing with a level matter since there's a lot to remember working it out can be a preferable alternative what is the smallest this can be the smallest it can be is equal to zero well minus three times zero be zero plus twenty that's my y coordinate and how would I make this equal to zero well X would have to be two that must be my x coordinate done Daisy so again Link in the description for this worksheet I would recommend attempting it again in a day or two just to see how much you know keep in mind it's only like five questions right I don't I don't want to overload you with questions if you want to find more questions you can do all sorts these are from the past papers though the year 12 papers not the year 13 ones not the a2s although I can use this as well the reason why I didn't take the year 13 ones is simple because there's a lot of extra stuff in there even in terms of um like you do complete the square and then there's like five parts after it so I didn't think it was worth it but I hope you enjoyed if you have any questions again leave them in the comments and I hope you have a wonderful day