okay so welcome to this video looking at the asp series on quadratics now there's a lot in this video so it's quite a long one so sorry about that but i would have been doing you a disservice if i was to shorten any of these or reduce some of the content for some of these questions so it's quite a long one but there's a lot of content to go through as you can see on the screen these are all the things that we're going to look at and there is an awful lot that is crossover from gcse and obviously a lot that then builds into this as pure module so we're going to be looking at quadratics but before we do just want to show you obviously to make sure how to use this video so let's just have a quick look so when you're on one of these videos obviously the screen will look like this and if you click on the video to pause it you'll have a selection uh all that little button down there that allows you to look at the chapters so if you go into the chapters you'll see on the side or on your computer it might come up below just to the side here it lists all the different topics that you can go through so if i click on expanding three binomials for this last video it then brings up the topic of expanding three binomials if you then click into the description which pops up on the right here you'll see that there is a revision video where it is bookmarked to the side so if you want to do or if you want to go further into expanding three binomials for example you can click the link and it will take you to a full lesson on that topic so hopefully that's really helpful and just explains how to use these videos in a little bit more depth so again these are the topics that we're going to be looking at hopefully that little explanation was helpful again please do like the video please share it with your friends and obviously please leave a comment on any content you'd like to see in the future but with that being said let's get started [Music] [Music] okay so this question here we've got a 2x squared so again i'll link in the description the the video for obviously factorizing harder quadratics so we'll explain them a little bit but i won't go into as much depth as i did in that video so do check that one out if you're not sure on how to factorize these now in terms of a 2x squared that means in one of our brackets here we're going to have to have a 2x when we're factorizing it so if i set the brackets up we've got a 2x and an x it's all equal to 0 and we just need to figure out where the numbers go now if we've got a 1 at the end our only factors can be one and one but one of them is going to be doubled okay obviously when we expand our double bracket here we've got a 2x and it's going to double that one in in the other bracket so as they're both ones we know they're both going to be ones here we just need to figure out the symbols now we want minus one x or minus one in the middle so in order to make minus one we're gonna want plus positive one take away two so as this one over here is going to be doubled we'll make that one become our minus two and this will be our positive one here and obviously you can go about expand uh expanding that just to check that it matches but there we go that is gonna match we're gonna get plus one on this expansion here plus one x and we're gonna get minus two x when we expand that one there that'll give us minus one in total so in terms of obviously getting our solutions here it's very similar to what we just looked at if we just set them both equal to zero this one here is just going to become x equals one obviously we haven't got a coefficient there and this one here we're gonna flip the sign so x is gonna equal minus one but then we're gonna divide it by the two so minus one over two or minus a half and there's our two solutions x is minus a half and x equals one and again you could obviously draw your sketch view quadratic there just think about what it looks like it's going to go through one and minus a half and you could draw that in there we go one and minus a half and that's where our two solutions are now in this one the process isn't any different but we've got something that's going to change here i just want to show you this and it's negative 11 here in front of x so if i write down the values of a b and c we have a equals 3 b equals negative 11 and c equals negative 13. and if i go ahead plugging all these values in i've got to have a look at what changes here so the start of our quadratic formula is minus b but b is already negative so if b is already negative that's going to turn into positive 11. so on the top there we have positive 11. i don't have to put the plus with it there but i just want to emphasize for you it's gonna be plus 11. okay so minus b if b is already negative that's going to become a positive number there so we have 11 plus or minus the square root of and again i've got to be careful here because i've got to do b squared now when i'm squaring a negative number and subbing a negative number in i should always really put it in brackets so negative 11 squared reason behind that and hopefully you've got your calculator in your hand if you type in negative 11 squared without the brackets you're gonna get negative 121. and a negative times a negative makes it positive so your calculator is not going to do it in the correct order for you it's going to do the um the power before it does the the minus okay so you just got to be very careful it does the order of operations so if you want to make sure it squares a negative number stick it in a bracket that's very very important when b is negative so we've got b squared minus 4 ac so 4 times 3 times negative 13 okay let's just extend that there we go that's all over 2a to all over 2 times 3. again on this question look that and that was new this time the minus b turning positive and then negative b in the bracket there when you're squaring it so being very very careful with that but there are new bits that you need to be careful on this type of question so let's type this in again we'll do the positive and we'll do the negative there we go and we'll stick these values in so fraction button positive 11 so just 11 plus the square root of in brackets negative 11 squared take away 4 times 3 times negative 13 all over 6 or 2 times 3. press equals and turn into a decimal you've got four point six zero seven two one nine four nine six and again finishing that off by rounding it to three significant figures here again we've got four point six one there we go and there's our first solution moving on to our next one back into the calculator takes a little bit of a while to click back there we go top it for a minus and we get minus one point five eight double six zero six eight seven five there we go and then finishing this off rounding it again that becomes negative one point five nine because it's a six there after the eight and there's our two solutions so when it comes to this question here obviously this is what we're going to complete the square for and we're going to find the turning point to start with and have a look at how we apply it for that as well so when we've got this one here now this is our quadratic that we have at the moment 2x squared plus 16x plus 26. now we can only complete the square well i'll say so we can only we we have to get the coefficient of x squared to be one first before going about completing the square so what we're going to do is we're going to factorize it by a factor of 2 okay so as they all divide by 2 there or just because x squared has a 2 in front of it we're going to take a factor of 2 out so if i factorize this by 2 i get 2 lots of and then dividing everything by 2 we get the 1x squared we get 8x and we get 13 at the end there we go now what i'm going to do is i'm going to basically ignore that that 2 is at the front so all i'm going to do is factorize or complete the square for what's in the middle here okay obviously we've factorized it there we've taken out a factor of two and now we're going to complete the square for what's inside so normal process for completing the square i am going to halve the coefficient of x so in my bracket i'm going to have x plus 4 obviously halving the 8 there and that is going to be squared obviously remembering why we do that because when we expand that we get the x squared we get the 8x but at the end if we expand it we'd get plus 16. now we don't want plus 16 we want the plus 13 there so we're going to have to take away 3 to make this balance out so i will have to put minus three at the end and now if i did expand and simplify all of that i would get the x squared plus eight x i'd get plus sixteen take away the three and that'll leave me with the thirteen that i'm looking for now obviously we can't forget that that two exists that two is still at the front there and we have to bring that two back in now i'm going to do is i'm going to put all of this in a bracket and rather than using a rounded bracket i'm going to put it in a square bracket just to define obviously it's a different set of brackets there and i'm just going to put that two back at the front all right there we go well we're almost done all i've got to do now is reintroduce that two and that just means expanding out this square bracket by that two and that's easy enough to do we've got two lots of the bracket there so i'm just going to write this out let's see what we've got we've got 2 lots of the x plus 4 squared there we go and then timesing that negative 3 by 2 will leave us with negative 6. and there we go that's that completed in completed square form all we have to do now is obviously like before we identify the turning point from what's in the bracket there so just like before the x coordinate is right here but we flip the sign so that is going to be negative 4 as our x coordinate and our y-coordinate is just at the end there the minus six so there we go our y-coordinate is minus six there we go and that's the coordinates of our turning point so very very similar to obviously normal completing the square but obviously we just have this extra element there of factorizing out that coefficient of x squared and then reintroducing at the end and just multiplying that number at the end there that what was normally our y-intercept um our own sorry our y-coordinate and actually just multiplying that back out by the factor that we took out at the start so this question we've got two x squared at the start okay and obviously when we're completing the square here we do have to take him to take that into account and we are gonna first complete the square it then says hence solve the equation so just like the last one i'm giving your answer in the form a plus b root three where a and b are integers so let's start with completing the square so we'll take out that factor of two so we have two lots of x squared minus four x minus eight okay and then let's complete the square for what's inside i'm gonna swap colors like i've done before there we go so half that coefficient of x and we have x minus two in bracket squared and that's going to make plus 4 at the end and we want minus 8 so we're going to have to subtract 12 there to make that balance out reintroducing that 2. let's see what we get there we go reintroduce the 2 and we have 2 lots of x minus 2 squared minus 24. now for the second part of this question here it says hence solve the equation so we're going to do exactly the same thing we're going to set it equal to zero so if i set this equal to zero and i'll do it separately up here we have two lots of x minus two squared minus twenty four equals zero now it's completely up to you in terms of the steps that you do here but there's a couple of things that we can get rid of from the start either we can get rid of this two at the start we could divide everything by two or we can add this 24 over to the other side now personally i just like to add the 24 over first but it's completely up to you in terms of which order that you do it i'll show you why it doesn't really matter but if i add the 24 over let's have a look we get 2 lots of x minus 2 squared equals 24. now at this point i'm going to divide by 2. so that's going to turn that 24 into a 12. now you could have divided by two at the start it wouldn't obviously affect the zero over here zero would still be zero but you've just got to remember if you do divide by that number you also have to divide the 24 by two because that's going to become 12 on the other side and you've got to make sure you divide everything by two on that left-hand side of the equal sign so personally i just like to add it over first i feel like it just helps you not to forget to also divide that by two as well so if we go about dividing by two now we can get we'll get x minus 2 squared equals 12 and then it's exactly the same as the last question now we can square root both sides and we get x minus 2 equals plus and minus the square root of 12 and then we can add that 2 over to finish it off and if we add the 2 over let's see we run out of space a little bit here but we get x equals 2 plus or minus the square root of 12. obviously we do just need to have a look and let's just bring this up here we do need to have a look at obviously writing in the form it wants and it wants it as b root 3. so we need to think obviously about that root there and that is root 12 and how does that simplify well let's have a look root 12 the square number that goes into that is 4 so we can write that as the square root of 4 times by the square root of 3 and the square root of 4 is 2 so that's 2 root 3. so finishing this off then we can write it in the form it's asking for and that is the x equals 2 plus and minus 2 root 3 and there is our final answer writing it in that third form that it's asking for in this question now just a very quick discussion of this before you maybe try and have a go but you will notice that the coefficient of x for one is an odd number so when completing the square that's going to form a decimal although you can use a calculator for this question but there is going to be decimals involved and a little hint for you this one doesn't factorize and when a quadratic doesn't factorize you need to use the quadratic formula so this is a calculator question the rest of those could have been non-calculator but this one here is explicitly calculator so a couple of little hints there but i would encourage you to have a go this is our last little question so if you pause the video and have a go otherwise let's have a go going through this one so if i go about completing the square to start with it wants us to draw a sketch showing the coordinates of the turning point and any intersections with the axes so let's take out that factor of minus one so when we do that we get x squared plus three x minus five now if we complete the square for this it's not too difficult particularly as we have a calculator but we get x plus 1.5 or you could leave it as a fraction 3 over 2 squared now when you expand that bracket 1.5 squared is 2.25 and we need to get back to negative 5. so you can do that on the calculator if you want if you need to but we're going from 2.25 down to zero and then an extra five so we're gonna have to take away seven point two five to get down to minus five now we need to multiply that minus one back in just like we have done before to put that minus 1 back on the outside and we get minus 1 lots of the x plus 1.5 squared minus 7.25 and obviously when we times that minus 1 in it becomes plus 7.25 so there we go we've got the coordinates of the turning point not too bad there hopefully minus 1.5 the opposite in the bracket and then 7.25 for our y-coordinate and again we can solve it from here and if we go about doing that um let's see what we're gonna get now we could solve it from there but we should really just solve it from the original quadratic once we've identified the fact that it doesn't factorize we may as well just use the original quadratic because it's not really going to matter which one we use there so if we bring the original quadratic down and we've solved from there we've got our values of a b and c that we're going to have to put into the quadratic formula so a is the number in front of x squared which is minus one b is the number in front of x which is minus three and c is the number at the end or at the start in this case which is the five and if we plug all that into the quadratic formula so obviously you need to know the quadratic formula for this as well so i'll put that in the description don't worry everything that's going on in this will be linked in the description for you to have a go out i'll try and put them in order for you as well for how they've occurred in the video but if we put those into the quadratic formula let's have a look so we have minus b at the start so that's positive 3 now when we flip that plus and minus the square root of b squared so negative 3 in brackets squared take away 4 times negative 1 times 5 and that's all over 2a or minus 2. 2 lots of minus 1. now if we type that into the calculator obviously using the fraction button there and just being careful how you sub all these numbers in using the negative button for your negative one putting your negative number in bracket there for the b squared for the version with the plus sign we get and i'm gonna round these to two decimal places for this one it doesn't actually say two in this question but for the purpose of our sketch we're not going to want to write all these decimals it comes out as minus 4.192 decimal places and if i go back into my calculator change that plus for a takeaway version with the takeaway comes out as and again to two decimal places 1.19 so there we go um obviously in a question like this we would just leave it as a full decimal if it didn't say how to round it but for the purpose of this little bit of practice we're not going to worry too much about that so obviously we've got a negative and a positive solution again so if we draw our quadratic again we're just going to do a little sketch and it's going to go let's have a look minus four over to one i'm trying to make it as accurate as i can or not not like over egging the uh whereabouts is on the graph too much but there we go we've got minus 4.19 over here we've got positive 1.19 we've got a y-intercept which again we can get from our original equation of five and we have our turning point which is just here not the best drawing there by me but minus 1.5 7.25 just about get that on there okay obviously just emphasizing the fact that that turning point is to the left of the y axis as we have a negative 1.5 as the x-coordinate so there we go really tiny little sketch there at the bottom but just showing you how this could obviously get a little bit harder obviously with completing the square with an odd number as the coefficient of x but if you have a calculator that's nice and easy enough and then obviously using the quadratic formula as well so it was a pretty tough question to finish on now the discriminant is something that you have met before uh usually obviously from gcse maths you'll have met the quadratic formula and it's something to do with something within the quadratic formula so if we have a look and just have a quick think about the quadratic formula which is minus b plus and minus the square root of b squared minus 4ac all over 2a now normally particularly at gcse level when you've looked at the quadratic formula it's very rare that anything other than a positive number comes up underneath that square root although you will have noticed before maybe you've typed it incorrectly sometimes and you may have found that obviously sometimes it came up with a maths error on your calculator when you type in something under here and you've got to be very careful with the numbers that you type into there um but obviously whether depending on whether it's a certain type of number depends on whether we get an answer on our calculator or we don't and that's what we're going to have a look at and actually understanding this bit now that bit on its own underneath the square root that b squared minus 4ac is what we call a discriminant and it determines when we have a quadratic how many solutions we have and whether we even have a solution at all and we're going to have a look at why and how that happens and a few particular questions here and the differences between them but essentially if we forget about the rest of the quadratic formula for the moment we are just concerned with this bit this b squared minus four ac and that is what we call the discriminant okay so we're just going to be having a look at this bit um so i'm just going to write that up up here we've got b squared minus 4ac and that is called our discriminant now so grab a piece of paper grab a pen make some notes we're going to get started with these questions here and just have a look at understanding why and how we get our solutions from this little bit under the square root now if you think about the concept of numbers underneath a square root okay we can have three different types of numbers we can have a whole number and let's pick something that actually um square root something like 25 and the square root of 25 would give us two solutions we can get plus five and we can get minus five so we get two solutions when there is a whole number under there sometimes it'll be an actual whole number sometimes it might be a decimal okay but we always get two solutions with a whole number we can also have the number zero and if we get the number zero obviously the square root of zero is zero but thinking about the rest of the quadratic formula that would mean we'd only get one solution because obviously in front of the square root there you have the plus and minus and if both the plus and the minus are both zero then both of our solutions are going to be equal or we could have and thinking about when this arises you could get a negative number under there so something like negative three let's just put something random in and you will get no answer on your calculator it would normally say calculator error and that's because we'll get no solutions or in terms of a quadratic when we're solving that we would call that no roots or no real roots okay no roots there we go i'll put that in there so thinking obviously about the concept of the quadratic formula and obviously i will link the video for that in the description if you need to touch up on the quadratic formula but in terms of the type of numbers that go underneath that square root will determine what answers we get when we look at these solutions and that's what we're going to have a look at with some of these questions so this first one that i've got on the screen i'm just going to get rid of all of this is looking at a particular quadratic and i have put a picture of this quadratic up here and we'll see obviously that we get these two roots or two solutions here on the x-axis so the equation for this particular quadratic is up here x squared plus four x plus two and in terms of actually figuring out the discriminant of that and we're going to work out the value of the discriminant we just need to plug in the values of a b and c now a is the number in front of x squared hopefully already happy with that and a in this particular case is 1. b is the number in front of x so b is positive 4 and c is that number at the end and c is 2. and if we stick all of these numbers into the discriminant so b squared minus 4ac let's see what we get so we've got 4 squared our value of b take away and then we have 4 times a which is 1 times c which is 2. and if we just work out this calculation here 4 squared is 16 take away 4 times 1 times 2 which is 8. we get 16 take away 8 and we get the answer eight so the value of our discriminant is eight and if we think about that if it was put into the quadratic formula we would have plus and minus root eight which would give us our two values underneath the square root there again that would give us our two solutions and ultimately determine these two solutions okay so in terms of stating whether this uh obviously without the graph has two roots two equal real roots which we'll discuss in a second no real roots we know that obviously with the positive 8 there with the plus and minus square root of 8 we would have two roots and we call those two distinct roots okay but i'm just going to put two roots here for this particular example and we can obviously see that from the graph as well but there's two elements there we've got the value of the discriminant and the fact that it has two roots which obviously is also shown on the graph so this question it says 4x minus 5 minus x squared also equals or is equal to q minus x plus p squared where p q and p are integers so obviously that there is highlighting the fact that it can be written in completed square form like that which looks ever slightly different to the way that we've been writing it in the earlier questions but we're going to have a look at how to actually write it like that so it says find the values of p and the values of q and then it starts to ask us some other little bits it says calculate the discriminant and then it also says sketch the curve showing any points of intersection so we'll have a look at those points step by step to start with we're going to have a look at obviously finding the values of p and q and writing it in this completed square form just here so let's write it out so we've got negative x squared and obviously this is a bonus because we've already discussed how to complete the square for this okay but let's have a look minus five so if we take out that factor of minus one to complete the square we'll get positive x squared minus four x plus five and completing the square for that let's obviously have that coefficient of x so we get x minus two in bracket squared and when we expand that we get plus four we want plus five so we need to add in an extra plus one there and then again not forgetting to reintroduce that minus one because that affects that y coordinate at the end there so minus one and we end up with minus 1 lots of x minus 2 squared and then that turns out into minus 1 at the end now obviously it wants it written in this form up here okay q take away those so all that means is putting one at the start there okay and just shifting this minus one from the end and putting it here at the start and if i write it out like that we have minus one take away the one lots of that bracket so i can get rid of that one there you go and there it is minus one take away x minus two in bracket squared so now it's in the format that we want we can write out what the value of p and what the value of q is so q is at the start so q is the minus one there at the start and p is negative two inside the bracket there we go and i always dislike that when they write it like this look they've written plus p just here and actually p's come out as a negative there p is negative two so do just watch out for that just because they put plus p there doesn't necessarily mean it's going to come out as a plus or a positive number so the next bit right it says calculate discriminant of this okay now when we're looking at the discriminant and again i've linked the video for that in the description we're looking at the value underneath the square root in the quadratic formula that b squared minus 4ac which is called the discriminant and it determines for us how many roots this is going to have so rather than getting us to actually factorize this if it does factorize and finding out whether there are two distinct real roots to equal real roots or no real roots we're actually working out the discriminants okay so if we put these values in then let's see what we've got so obviously um we need to get the values of a b and c from here now i'm going to take it instead from up here where i've rearranged it so the value of a is that one in front of x squared there so a equals the negative 1 in front of the x squared we've got b which is in front of the x so the positive 4. so b equals four and then we've gotta forget c as well which is the number at the end which is minus five so there we go c equals minus five and if we sub all these values into b squared minus four ac let's see what we get we get four squared take away four times minus one times minus five there we go and if we work that out let's have a look just multiplying all these bits together to start with four times minus one is minus four times minus five is positive twenty so we have sixteen take away positive twenty or sixteen take twenty which is minus four so my discriminant has a value of minus four now obviously what we know about the discriminant obviously hopefully you've already checked that video out and you know about the discriminant if we get a negative number that means that means we have no real roots so in terms of this uh quadratic that we're going to draw it's not going to cross through the x-axis and that's what that tells us and obviously if we were asked to draw this curve and we weren't asked to calculate the discriminant we might be sitting there trying to factorize it we might be putting it into a calculator trying to get in quadratic formula but actually very quickly we can tell using the discriminant whether it does actually have any roots so it's a nice quick way for us to figure that out and obviously this question asks us to do it anyway but just thinking about another type of question if it didn't ask us to calculate the discriminant uh you know it would be a nice quick way just for us to see whether it does factorize or not and that's quite nice about the discriminants as well so if we go about actually drawing this if we want to sketch this curve showing clearly any points of intersection with the axes let's have a think about what it would look like we know that it's going to be an upside down or an n shaped quadratic curve here so if i just draw a little basic axes one thing we do need to get is obviously that y-intercept which we've got up here the y-intercept is minus five so that's going to be down here somewhere so i probably want to draw this a little bit better actually let's get rid of that let's draw it so we can get actually down the bottom there so we know it's going to be -5 we know it doesn't even reach the x-axis there we go and the only thing we need to figure out is whereabouts we're going to put it now we can we can think about where the maximum point is because we've completed the square up here so we've completed the square and we have got from that just here we've got positive two and minus one as our turning point and if we go about actually finding that let's imagine two's there minus one is there so it's around here there we go so if i try and draw this in then we've got it crosses through at minus five we've got a maximum point there and it never touches the x axis and obviously just labeling this on we've got minus five there and that's all we need to label it doesn't ask us to label the maximum point there we could put it on if we wanted but there we go there is a sketch of what this curve actually looks like there we got the minus 5 the y intercept up from our equation up here okay well let's write that in y equals minus five there we go and we've got the fact of where the turning point was going to be positioned just here from completing the square again and then using the discriminant we figured out there was going to be no roots so we know it wasn't going to get up to the x-axis there so there we go that's how we go about all of these points obviously completing the square to find the turning point to get an idea of whereabouts is going to be positioned obviously then getting the y intercept from our equation and using the discriminant to figure out whether it was going to touch the x-axis or not now i would encourage you to obviously just pause the video have a read of this question before we start going through it because there's a lot of information a lot of questions and a lot of things are probably just worth reading before we actually get started so i'll give you an opportunity to do that now okay so a javelin is thrown over level ground from the top of a tower and the height in meters of the javelin above the ground after t seconds is modeled by this function so we've got the function here for t and it says 12.25 plus 14.7 t minus 4.9 t squared where t is greater than zero there we go so it says here interpret a meaning of the constant and then it starts to give us some more questions so what we're actually going to have a look at is just thinking about what this would look like in terms of a function in terms of a quadratic and sketching that and then we'll have a look at answering all of these little questions now before we do that i'm just going to make this a little bit smaller so hopefully maybe you've made some notes on that you've written it down because this is going to be a little bit harder to read because we need quite a bit of space so i'm going to push that to the side and we're going to carry on looking it from here so if we were going to draw a little sketch of this and we kind of imagine what this would look like on a graph so if we draw a basic little axis and it does only have to be a little sketch here you don't really even have to draw this so it does help us just to visualize it so we're standing on the top of a tower and let's imagine that's on the y-axis there and we're throwing a javelin let's just say this is the top of our tower and we're going to throw that javelin and let's go for a different color to the right and that is going to make a nice quadratic curve and it's going to land somewhere down there now obviously if we were going to model the entire quadratic the actual curve itself would continue but obviously we know here we're throwing it forwards off the tower so we don't really need to know about that but it does help us just to kind of visualize what's going on now if we look at the actual um function here we've been given a load of numbers now we should already know from sketching quadratics and just knowing about coordinate geometry and our line equations that one of those numbers there is our y-intercept and it's the constant number so not the t-squared or the t but the 12.25 so that 12.25 there in our function indicates where it crosses through the y axis and in the case of this question that is actually indicating the height at which it is thrown from so that's 12.25 now straight away that actually answers part a for us because it says here interpret the meaning of the constant term 12.25 in the model well that is the height of the tower or in other words the height at which the javelin is going to be thrown from and in this case that is the height of the tower so that's the answer to part a it's going to be the height of the tower let's have a look at part b it says after how many seconds does the javelin hit the ground now obviously if we're looking at how many seconds it takes for javelin to hit the ground we first needed to think about the axes here we already know that 12.25 in terms of this diagram is our height so we can label that axis as h and our other part of the axis is going to be our time and obviously we could label that as t which is given as in the question and that time is in seconds so if we want to find where the javelin hits the ground we're looking at specific points on this quadratic and specifically we're looking at the solutions these points here and obviously this point here if we were to go backwards they're the two key points but we're obviously looking for that positive one if we're throwing the javelin there to the right so in order to find the solutions what topic do we actually need to know we need to know obviously how to solve quadratic so that could be down to either factorizing to solve it or it could be the quadratic formula now look at the numbers that are given to us in this function it's going to be a lot simpler for us to use the quadratic formula particularly as you would have a calculator for this question so really in order to find that number we just need to type these numbers into the formula so first of all we just want to write down what the values of a b and c are in order to put that in obviously if you have some of the fancier calculators you can actually just type those numbers and it will give you the solutions but we should know how to use the quadratic formula and i'm not going to assume that everybody has one of those calculators where you can do that so for part b here let's just write down those values so the value of a is the coefficient of t squared which in this case and if we highlight it is negative 4.9 so our value of a is negative 4.9 our value of b is going to be the positive 14.7 in the middle there so b is 14.7 our value of c is the constant at the end which is actually the start of ours but 12.25 so at this point obviously you do need to know the quadratic formula so again i'll link that in the video and if you need to have a look over the quadratic formula please do feel free to do so and then come back to this so plugging this into our formula then so we've got minus b so minus 14.7 plus and minus the square root of b squared now b is not negative so i won't worry about putting it into a bracket but 14.7 squared and then minus four ac we have to extend that because we've got quite a lot going on here so 4 times a a is negative 4.9 so times negative 4.9 times c which is 12.25 there we go just about fit all of that on the screen and that's all over 2a so all over 2a a is minus 4.9 so 2 times negative 4.9 so typing that all into the calculator once with the plus sign and if you type that in with your plus sign so i'll just indicate that by putting a little plus there we get the answer negative 0.679 now obviously we know it's not going to be minus 0.679 seconds so we're going to have to go back into the formula using the minus sign and let's do that now so back into the calculator take away the plus put in your minus and we get here 3.679 so there is some more decimals there i'm going to leave that as three decimal places or i could say four significant figures there we go and again you could leave it as two or three significant figures but obviously just write down the way that you're rounding it there particularly when it doesn't ask you how to round it in the question so there we go that's going to be part b for us it has to be that positive value there so our value is going to be 3.679 so you've already seen with modeling quadratics there's already one extra or additional topic within this that we're going to need to use in order to solve them if it mean if we are in a question where we do need to find one of those solutions where it's crossing over the axes so that is part b we have found after how many seconds does the japanese hit the ground well after 3.679 seconds and that's the answer to part b now we're obviously going to have a look at part c so let's get rid of some of this working out because we're not going to need all of that we'll keep the answer on the screen but we'll get rid of all this quadratic formula and everything and we'll just keep a fresh eye for this next question there we go right let's just move this up a little bit there we go and we can say that that there was part b obviously don't rub your working outing when you're doing this but for the purpose of fitting all this on the screen we'll go from there okay so for part c it says write the function of t here in the form a minus b brackets t minus c squared where a b and c are constants to be found so hopefully you are recognizing that there and that is completed square format so we do have to complete the square for this quadratic and obviously that involves you being able to complete the square for harder quadratics and in this particular case you need to complete the square for a negative quadratic and i mentioned that at the start of the video and if you're not entirely sure on completing the square when we're doing this for a negative quadratic again please do go into the description check that topic out and then come back here and make sure that you're okay with obviously completing the square for a negative quadratic so in order to do this i think the easiest way and there are obviously some slightly different ways of approaching completing the square but i'm just going to rearrange this quadratic so that the t squared is at the start so if we just move that piece to the front that would be minus 4.9 t squared we've got the plus 14.7 t in the middle and then we've got a positive 12.25 at the end remember we're not rearranging the like uh rearranging it to the other side of that equal sign we're literally just moving them into a position that looks like a normal quadratic so from this point here we obviously need to take the coefficient of t squared out and in that case this case that's for the negative 4.9 so we're going to divide everything by negative 4.9 now don't forget you've got a calculator so although the numbers don't look very nice it's relatively simple to do that you just need to divide all these pieces by the negative 4.9 and when we write that we put the minus 4.9 on the outside open up a bracket and then we'll divide everything now so we'll get t squared and again you're just typing this in on the calculator feel free to type it in along with me 14.7 divided by negative 4.9 will come out as -3 so minus 3t and then 12.25 divided by negative 4.9 comes out as negative 2.5 and there we go our quadratic is now in a position where we can complete the square so opening up a new set of brackets so we can just complete the square in the middle we've still got the negative 4.9 and again there are different ways of approaching this next step i like to just complete the square as we always did at gcse level so completing the square halving the coefficient of t there would give us minus 1.5 in bracket squared if we expanded that bracket that would give us positive 2.25 we don't want positive 2.25 we want negative 2.5 so in order to complete that squared bracket i would have to subtract 4.75 and again you can work that out on a calculator hopefully you can spot that though that to get from positive 2.25 back to negative 2.5 we have to take away 4.75 and there we go i can close my bracket right so at this point again we're running out of a bit of space so i'm going to get rid of that first bit of working out and we're just going to go from this position that we're in at the moment there we go so at the moment we have these particular pieces we've got the minus 4.9 on the outside and we're going to have to multiply that back in so that we can have a look at those particular elements so let's multiply the negative 4.9 back in which just means it goes at the start of the squared bracket so minus 4.9 open brackets t minus 1.5 squared and then obviously we need to multiply that constant at the end by the negative 4.9 so negative 4.75 on your calculator multiplied by negative 4.9 gives us positive or plus 23 point so there we go we've completed the square not a nice one but obviously do check out factor completing the square and my negative quadratics video if you're a little bit unsure on that obviously in the question again let's have a look at this let's just highlight that again in the question it did ask us to write it in a particular form and it wanted it as a minus and then the complete square bracket there so we just need to obviously slightly rearrange this we're going to move that 23.275 to the start just before the negative 4.9 and that would then read 23.275 take away 4.9 brackets t minus 1.5 squared and there we go and that is in the form that the questions asked us to do and we've completed the square and we've completed that question now you might be asked for obviously what the constants a b and c are and if you are you've got them right there you've got a is 23.27 b is 4.9 obviously because minus the negative is already in the question there and c is 1.5 again the negative is already in the uh the way that it's written so c is just 1.5 right there we go so that is part c again for the next question for part d we're not going to need all the working out here so let's just get rid of that working out again hopefully you're writing this down as you're following along and there we go get rid of all those little pieces and there is part b sorry part c there we go move that up right onto the last part then part d and the good thing about these questions is part d is usually always using one of the previous parts and this one actually tells you which part it wants you to use so it says using your answer from part c or otherwise find the maximum height of the javelin above the ground and the time at which the maximum height is reached now in order to understand this you just need to understand turning points of a quadratic and what actually completing the square finds for you if we think back to our diagram the turning point is the highest point this point here on the top of the curve where it stops going up and it starts going down and that is found by reading the numbers inside our completed square form and specifically we read the number in the bracket for our x-coordinate and the number at the end the constant there for our y-coordinate so if we actually wanted to label this as a pair of coordinates not forgetting that the number in the bracket the sign is going to change there so it's not going to be negative 1.5 but it'll be positive 1.5 so this turning point coordinate would be 1.5 and 23.275 again if you've forgotten that i'll stick that link in the description for completing the square as well just a little reminder from gcse level so we've got all of our numbers let's see how we apply that so it says using your answer from part c or otherwise find the maximum height of the javelin while the height is our y-coordinate so the height there i'm just going to write this as height and this is question d so for the height is our y coordinate which is 23.275 and the question is in meters so i would say 23.275 meters for the next part it has also asked us to find and the time at which the maximum height is reached and the time there is our x-coordinate as you can see on the x-axis that was our time and that is the 1.5 so for the time it's given to us in our coordinate there and that is 1.5 looking at the units in the question the units are seconds and there we go and there's our question completed so as you can see there was a lot of topics involved there we had sketching the quadratic not that that was essential but it does help just to visualize the question so sketching a quadratic we had using the quadratic formula we had actually in part a reading the y-intercept and understanding what that meant we had completing the square and doing that with a negative quadratic in this question and obviously interpreting that and looking at the turning point and there we go that is the end of the video so we've covered all of these topics there's an awful lot in there don't forget as i showed you at the start to go into the video and check out some of those additional lessons if you feel that you need to on any of these topics so hopefully that was useful and helpful if it was again please don't forget to like the video share it with your friends and obviously leave a comment for anything that you'd like to have a look at in the future if any there's any links i could maybe post to you that would be helpful but there we go see you on the next one [Music] [Music] oh