today we're starting chapter two uh we're going to start with quadratics okay and what we're going to have is we're going to have a set of skills that we need to be able to master and this is the absolute minimum requirement of these skills and then we have our success criteria abstract now you all have a copy of this you're able to go back and refer to it at any time so what we're looking at in chapter two is we're going to be looking at solving quadratic equations completing the square looking at quadratics as functions quadratic graphs and the discriminant so the discriminant is something completely new okay but everything else is an extension or is gcse level so um obviously why do we use quadratics we use them for a lot of things but projectile motion is probably the best one looking at um distance and displacement you know forecasting what's going to happen what curve is going to look like and we're going to have a bit more of a discussion about that once we get through the um chatter so i would like you all as a starter to quickly solve by factorizing and solve by using the quadratic formula off you go make sure it is equal to zero so we're gonna minus six two numbers that multiply to give me minus six but add to give me five negative three no plus six negative one okay and that equals zero but when we're solving quadratics okay this isn't just the answer we've just factorized it what number would make this bracket zero okay and it's when x is minus six i know some of us like to say we change the sum but we really need to understand what's going on because this is going to help us when we look at the discriminant and start sketching so because if you think i've got a bracket multiplied by a bracket equals zero so either this bracket needs to be zero for the c equals zero or this bracket needs to be zero that's why we change the sign and it's really important we understand why so x equals negative six x equals one okay using the quadratic formula we should be happy to use the quadratic formula and we have our abc so our a equals one b equals five c equals minus six we should know the quadratic formula off by heart minus b plus or minus the square root of 4 sorry b squared minus 4ac all over 2a okay it's really important we memorize this because we are going to need this for the discriminant we plug in we substitute in our values so wherever there's an a we're going to put a 1 wherever there's a b we're going to put a 5. it's really important that whenever we substitute values in we are writing brackets and you'll see what i mean in a second so then when you substitute it in you're gonna have minus bracket b it's so important we use brackets plus or minus the square root of five squared minus four lots of a which is one c which is minus 6 all over 2 bracket 1. we're going to get answers of minus 6 or 1. alternatively use your calculator go to the function that says equation and type in your a b and c and you will get those answers okay so if we are solving without factorizing a lot of people look at this x minus 1 squared equals 5 and immediately people want to expand that bracket there is no need to expand that bracket okay because how do you get rid of a square root i mean a square you square root it so we're going to square root both sides so we're going to get x minus 1 equals plus or minus the square root of five okay then i'm left with x minus one so i'm gonna add one to both sides so i'm gonna get x equals one plus or minus root five so i'm gonna have two values for x x equals 1 plus root 5 or x equals 1 minus root 5. now we like to keep things as numbers okay as exact numbers we wouldn't turn these into decimals we would keep them exact so we have quadratics in disguise so here i'm going to rewrite this we have x minus six to the power of a half plus eight equals zero now if we multiplied x to the power of a half times x to the power of a half i'm going to get x so this is in fact x to the power of a half squared so this is going to be x to the power of a half squared because that's going to give us x to the power of 1 minus 6x to the power of a half plus eight equals zero okay so i'm just all i've done is rewritten x in terms of halves x to the power of a half now what we can see here is x to the power of half something squared minus six lots of something plus a it is a quadratic in disguise because quadratics always go something squared plus something x plus a number so what we're going to do is we're going to let y equal x to the power of a half so wherever there's an x to the power of a half i'm going to substitute in y so now i'm going to have y squared minus 6 y plus 8 equals 0. now this is a form that we know once we've seen once we've just undisguised it we've substituted we all know how to solve this now the hardest part of this question is finding what we need to substitute this will come with practice now we have a quadratic we should be able to solve it minus four y equals minus two equals zero so now we get y equals four and y equals two as our answers but we haven't finished yet because we used substitution we have to go back we know we said that y equals x to the half okay so this is saying x to the power of a half equals four or x to the power of a half equals two we don't want x to the power of a half we want x so to get x we have to square both sides because that would give us x to the power of one so if i square both sides i'm going to get x equals 4 squared which is in fact 16 and x equals 2 squared which is in fact 4. so the answers for the question is 16 and 4. okay now this comes in why we're getting x on its own this comes in from indicy laws from previous chapter one okay and just to give a bit more clarification x to the half squared we know is x to the power of one and whatever we do to one side of the equation we have to do to the other side what i'd like you to do now is to test your understanding okay you have four questions to do depending on where you are you might choose to go for the green you might choose to go for yellow you might choose to go for red i'm completely happy wherever you start you decide how confident did you feel if you felt really confident go straight for red if you didn't feel really confident maybe try some of the um the green if you felt a bit oh okay go for orange give it a go please you know i said in the previous example it's easier for you to square root this only works if you have an integer on the right hand side because you're going to get a number so for example for this one i'd have 2x plus 1 equals plus or minus root 5. for the first one i would expand out the bracket for the first one i would expand out the bracket um and then start my working from there for the third one i'd square both sides so i get x plus three equals x minus three squared and i'd expand the bracket on the right hand side and then for the last one i'd use substitution let y equal x to the power of a half okay so for the first one i'm going to have x squared plus six x plus nine equals x plus five if i minus x and minus 5 from both sides i'm going to get x squared plus 5x plus 4 equals 0. factorize i'm going to get x plus 4 x plus 1 equals 0. so x equals negative four or x equals negative one for question two i'm gonna minus one from both sides to get minus one plus or minus root five then i'm gonna divide both sides by two to get minus one plus or minus root five all over two for question three i'm gonna get x plus three equals x minus three squared which is x minus three times x minus 3. so i'm going to get x squared minus 6x plus 9. i'm going to minus x and 3 from both sides so x squared minus 7x plus six factorize x minus six x minus one equals zero x equals six or x equals one and then question four we substitute in okay we have a quadratic that we're used to we solve the quadratic we get our two values for y we need to make sure we substitute back in to find our values for x so completing the square summary we should all be very happy expanding x plus nine we should have x squared plus 18 x plus 81 okay and x minus 5 we should have x squared minus 10x and plus 25. everybody should be happy with that okay and the relationship between the bold numbers is they are square numbers okay um when you think your plus nine and your minus five we square them sorry they're not square numbers but we square them to get this 81 and this 25 that's what the relationship is now when we're completing the square if you have x squared plus 12x we could say essentially this is the same as writing x plus now with the number 12 we halve it so we're going to get 6 squared minus 36 now the reason i've done sorry i'm going to put minus 6 squared the reason i've done that is because if i did x plus 6 squared i would get an extra 36 and i don't want that so i minus 36 okay so those of you who do not know how to complete the square again you need to know this and i'm just going to run through it really quickly it's always x plus or minus depending on the question something squared minus something squared everyone happy with that okay this is the rule so this number seven just goes at the end here so you can just write that there straight away okay whatever this number is you divide it by two and then you put it here okay very simple then whatever this number is that you've divided by 2 with the sign goes here so then we're going to have minus 3 plus 7. that is how we complete the square so we're going to have x minus 3 squared minus nine plus seven we're going to have x minus three squared minus two we should have remembered that from gcse okay if not please work on that at home so then uh those of you who are confident can you go straight to this question here those of you who are not who have seen that before i'm going to show you how to do this different one okay if you're confident go straight down to the bottom question now the method i just showed you where it was bracket bracket x plus squared minus bracket bracket squared it's always the same but this only works if you have a number one in front of the x squared so the coefficient of x squared is one that's what i mean the coefficient means the number in front of the x squared is one in this case the number is two so we cannot start with that formula okay we have to have it has to be one for this method up here to work so what we do is we divide everything by two so i'm going to take two out and put a square bracket okay i'm happy for you to all watch this one first if you want to 2x squared divided by two is just x squared twelve x divided by two is six x seven divided by two is three and a half but we love to keep things as fractions so we're gonna write seven over two okay and then i'm going to finish with my square bracket everybody happy now this 2 in the square bracket we're going to leave exactly how it is just leave it okay we're not going to worry about it for now because what we have here is x squared plus 6x plus 7 over the coefficient of x is now one we only have one x squared so i can use this formula that i've just shown you okay so i start off with bracket x plus remember i half this one what is half of six three bracket squared minus bracket whatever this is is free i put here squared plus seven over two because that was what was on the eight so now once we've done that um we can leave our two again on the outside but what we're gonna do is we're going to simplify what's inside the square bracket so we're going to have x plus 3 squared minus 9 plus 7 over 2 is 18 over 2 plus 7 over 2 minus 11 over 2 okay so i've simplified it so then what i'm going to do is i'm going to multiply 2 by the bracket which is going to give me 2 bracket x plus 3 squared and then i'm going to multiply 2 by minus 11 over 2 which is just minus 11. so that is the actual answer now i'm going to do the exact same for the second example this time when i look at the example instead of i need the only way i can use this formula x plus or minus squared minus bracket bracket squared the only way i can use that formula is if x has is just x squared on its own okay the coefficient of x is one here the coefficient is minus three so what do you think i'm going to take out the bracket minus three okay and then i'm gonna have square brackets and i'm gonna divide everything by minus 3. 5 divided by minus 3 is minus 5 over 3. minus 3x squared divided by minus 3 is plus x squared 6x divided by minus 3 minus 2x okay again a lot of this i'm doing in my head please feel free to use calculator if you need to i'm going to rewrite this minus 3 bracket and i'm going to write it in the form x squared minus 2x minus 5 over 3 and close bracket okay now i have it in the form i need it so i'm going to leave my minus 3 on the outside the coefficient of x squared is 1. so now i'm going to put it in completely the square form bracket x half of b minus 1 squared minus minus 1 squared minus 5 over 3. okay now what i'm going to have is minus 3 open bracket x minus one squared minus one squared is one minus one minus five over three becomes minus eight over three then i'm going to multiply everything by minus three so i'm gonna get minus three bracket x minus one squared plus eight now again a lot of the multiplication i'm doing in my head and we should be at the stage where we're comfortable doing this in our head giving you 10 minutes which is a lot more than really i would normally give so step one 3x squared minus 18 x plus 4. the coefficient in front of the x squared okay is a free okay so first of all i need to divide everything by three so i'm gonna get free bracket x squared minus eighteen divided by three is minus six x four divided by 3 is plus 4 over 3. now inside my square bracket i have x squared minus 6x plus 4 over 3. that now means i can complete the square as normal so i'm going to get free bracket x minus three squared minus three squared plus four over three okay which i'm then going to simplify inside the square bracket x minus three squared minus 9 27 minus 23 over 3. now i'm gonna multiply everything by three so i'm going to get free bracket x minus three squared minus 23. and that is the answer for the second part of the question in front the coefficient in front of the x squared is negative 5 so i need to take negative 5 out and divide everything by negative 5. x divided by negative five is negative four x negative five x squared divided by negative five is x squared three divided by negative five is negative three over five and now i'm going to rearrange this in the form x squared at the beginning i'm going to get negative 5 square bracket x squared minus 4 x minus 3 over 5. now it is in the form that i can complete the square so i'm going to get negative 5 open bracket x minus 2 squared minus minus 2 squared minus 3 over five okay minus two and then i'm going to get minus five bracket x minus two squared uh minus four times five minus four minus twenty is it minus 23 over five oh sorry no what wait there minus 4 is minus 20 over 5. i get the same again okay then i'm going to multiply everything by minus five so i'm going to get minus five x minus two squared plus twenty three i'm going to rewrite that so i've got 23 minus five lots of x minus two squared so okay we're going to do the exact same again now if we look at this question we'll notice the first question is the same question as the first question before so step one is complete the square which we've already done on the previous slide we're going to get free bracket x minus 3 squared minus 23. we've already done that on the previous slide everybody should be happy with that now it says solve the equation this is where it's a bit different solve the equation and here's the equation and it equals zero we have to solve this now the reason why we have to solve this equation by completing the square is because if you try it now try this into your calculator your a equals three your b equals minus 18 and your c equals four try that into your calculator using the solving the equation button and it will come up as error okay it turns out this one we can factorize normally we complete the square when we cannot factorize okay how we solve this though i'm so sorry about that is we just once we complete the square we have it in the form we have it in the form 3 logs of x minus 3 squared minus 23 equals 0. all we're going to do now is classic rearranging i'm going to add 23 to both sides so i'm going to 3 lots of x minus 3 squared equals 23. then i'm going to divide both sides by three so x minus three squared equals twenty three over three then to get rid of the squared i'm going to square root so i'm going to get x minus 3 equals plus or minus the square root of 23 over 3. to get rid of the minus 3 i'm going to add 3 to both sides so i'm going to get x equals 3 plus or minus the square root of 23 over 3. and that is the answer yeah so this is the final answer okay and that's how we've done it by solving what i would like you to do now is um i know we've only got a few minutes left of the lesson but everyone should be completing the doctor framework if you have struggled with this lesson you have extra homework to do you have to do question three a c and d extra homework a c and d as well as dr frost as well as the formal homework for chapter one and i will set something else okay and this is all due in on sunday what was the question faisal which is x squared plus 3x minus 1. x squared plus 3x minus 4. okay and how many marks did i give four marks let's see who's got four marks then step one it's a sketch okay you know it's a quadratic so you don't even have to worry about anything when you're sketching you know if we factorize we'll get the roots so i'm gonna get x plus four x minus one so i know this is going to be minus four and this is going to be one now that i know that i know one is roughly here minus four is roughly here but i don't actually put those dots in but i know i'm going to have more on the left hand side if that makes sense on the right the reason why i don't put those dots in is because i do this voila perfect this is one this is minus four okay this one is also minus four it's the y intercept we'll get that here okay so far you've got one mark for one mark for this one mark for the minus i'll do this in a different color one mark for minus four and one okay then the next mark you're going to get is you need to complete the square to find the minimum point so we're going to have x plus 3 over 2 squared minus 3 over 2 squares 9 over 4 minus 4 16 so minus 25 over 4. so we're going to get x plus 3 over 2 squared minus 2 5 over 4. now the minimum point is whatever's inside the brackets essentially we flip it but i don't like saying it like that when does x equal zero uh when does that bracket equal zero and that's when x is minus three over two and then the y is just um the integer that's left over so then this point here is your minimum point okay and this is minus 3 over 2 and minus 25 over 4. okay so there's three marks so far not yet najeeb hold fire okay this is one mark here are you ready for the last mark please make sure you label your x and your y axis okay so everybody always forgets that but that's an easy one mark okay okay so today we're moving on to the last part of our um chapter and we're going to get through this in the next 20 minutes we're going to go fast pace it is very simple um functions very easy you have an input you have an output so this function what we've done we've multiplied it by two because if we input x and we get out 2x it's a multiplied by two now a function is the same as a lot of equations okay so when we write in equation form these are functions x squared is a function you put in one value you get another value now something you need to write down the domain of a function is the set of possible inputs so the domain of a function is normally your x values okay so write that down domain is possible inputs which is normally you input so for here we'd input x's wouldn't we so domain is your x values and your range is the possible output now when we're looking at a graph your range is normally your y coordinate and again i'm going to explain that in a bit more detail in a minute okay so everyone should be happy now a function can only exist if it's one to one okay that means i put one number in and another number comes out okay but with a function i can have for example the same output so for example if i put in minus 4 i could get 16 because minus 4 squared is 16. if i put in 4 i also get 16. i can get the same output okay right then so this is um back to gcse stuff so we should be fairly confident with this we have a function x squared minus three x and g x equals x plus five x belongs to the real numbers so this is our domain this is telling us that x can be any number okay so we know x is any number on the graph so if we had a graph sometimes it tells us that x is between three and two but this it says any it belongs to real numbers so it's any number along here any number across the x axis we should be happy with that so number one find f is minus four so i'm going to substitute in minus four so if we think about a function this is f of x and everywhere there's an x we're going to substitute in a minus four when we use substitution we need to make sure we put brackets minus four squared minus three lots of minus four those of you who struggle with mental arithmetic in your head the reason why we do brackets is because when you put it into your calculator your calculator will do bid mass okay so we need to do brackets first so 16 plus 12. so f of minus 4 is 28 and again if you can't do that in your head please use a calculator but use your brackets find the roots of f of x when we're finding the roots that means we are solving it we put it equal to zero and we solve it if it says find the roots equal to zero and solve so we're going to get x squared minus three x equals zero in this case we can factorize x brackets x minus three equals zero now again how we find the roots when does this equal zero okay now either x has to equal zero because zero times bracket is zero or the bracket has to equal zero because zero times this number will be zero so we're gonna have two options x equals oh sorry x equals zero or the bracket which is x minus three equals zero if x minus three equals zero that means that x has to equal three so these are the roots x equals 0 and x equals 3. those are the roots of the equation and the roots mean where it passes on the x-axis here we go that is where the roots are then says find the roots of f of x at g of x so again when we're finding roots you put it equal to zero and solve it so x plus five equals zero so x must equal minus five nice and simple it's only got one root great ah sorry everyone i have i've done c and d as b and we've we've lost b i'll do b now okay so this is c this is d this is b okay so nice and easy for the next one right then we have b find the values of x which f of x equals g of x okay f of x equals g of x this should be nice and simple for you f of x is x squared minus three x so instead of writing f of x you're gonna write x squared minus three x equals and your g of x is x plus five so instead of g x g of x you're gonna write x plus 5. we're going to rearrange i'm going to put everything on the left-hand side x squared minus 3x minus x minus 4x minus 5 equals 0. i'm going to solve this okay so we're gonna get x minus five and x plus one equals zero again when are the bracket zero when x equals five or x equals negative one okay we should be fairly happy with that from gcses the next example is when it says find the minimum or the maximum and as we've just seen by sketching the graph to find the minimum or the maximum what do we need to do complete the square we we could do that and we're going to look into that but at the minute it's just about completing the square so very quickly x minus 3 squared minus minus three squared plus two we're going to get x minus three squared minus nine plus two is minus seven this equals zero okay so when we're finding our minimum value what makes the bracket zero three and then our y coordinate is minus seven when we're trying to solve for x that's when we put it into equal to zero but we can find our minimum point by putting it in this way and we're going to come to why it works in a moment so we're going to complete the square so we're going to get x plus 2 squared um minus 4 plus 5 minus 4 plus 9 is plus 5. so we're going to get minus 2 and 5 and we want to find x for which this minimum value oh right sorry one's asking for the y coordinate one's asking for the x i've just written the whole coordinate but really the minimum value of the function sorry i didn't read the question the minimum value of the function is five and the minimum value for x is minus two okay but i've just written it in coordinate form i'm happy for us to do coordinates right all we're doing is x squared plus four x plus nine we cannot find the minimum point without putting it in completed the square form this is completely equivalent okay to x plus two squared plus five and that's because if i expand this out x plus two x plus two plus five x plus times x x squared 2 times 2 4. 2 times x 2x 2 times x 2x plus 5. x squared plus 4x plus 9. this equals the left-hand side we are not changing what this is this is the exact same function we're just writing it a different way that's all we're doing for example the number four we can write that as two squared we can write that as five minus one as long as we do not change the value we can rewrite any mathematics any way we want okay so we can always double check if we're right or not by using that okay then next up so x minus five squared so minus 25 plus 21 minus 4 okay so our minimum values minus 4 and this occurs when x equals 5. so my i'm going to get 5 minus 4 is my coordinate for the minimum now when we have a negative x squared we don't have a minimum minimum we have a maximum okay so 10 minus x squared how am i going to solve that and find the minimum or the maximum okay realistically this is incompleted the square form already because we have no coefficient of x so if we were to complete the square of the square on this we'd end up with um a zero as with half zero is zero so this is the this is already incomplete of the square form essentially because we can't do anything about it so our coordinates for this is 0 10. okay because there's no coefficient in front of the x it's just how it is so then for the last one we have a minus x squared when i'm completing the square with a minus i need to take a minus out of everything so i'm going to have x squared minus 6x minus a now i can complete the square on that so i'm going to get x minus 3 squared um minus 9 minus 8 is minus 17 okay then i put the minus back in so i'm going to get minus x minus 3 squared minus 17 again so my maximum is at sorry plus 17 because i multiply out so my maximum value is 17 when x is free so my coordinate is 3 and 17 please make sure that we're happy with those okay so if you are happy you go straight to question free if you got all of them right straight to question three okay if you're not happy um well no i'm not gonna use the word not happy if you're less happy no if you're satisfied can you go to question one or two you can pick okay and if you have no idea what's going on i will be walking around the aisles shortly okay so for the first one we're finding the minimum value and if we're finding the minimum or the maximum we've got to complete the square so i'm going to get 2 lots of x squared plus 6 x minus five over two which is two lots of x plus three squared minus nine which is 18 over 2 minus 23 over 2. and then i'm going to get x plus 3 2 lots of x 3 squared minus 23 so my minimum value is going to be at minus 3 and minus 23 okay for the second one we're finding the roots of the equation so we're going to have to factorize so two x x plus one plus one this equals zero okay so when does this bracket equal zero well so two x plus one equals zero when x equals negative a half and this bracket x plus one equals zero when x equals negative one so these are the roots then for the last one we need to do substitution so if we let y equal x squared we're going to get y squared minus y minus 6 equals zero we're going to get y minus 3 and y plus 2 equals zero y equals positive three and y equals negative two now remember that we use the substitution so now x squared equals three and x squared equals negative two for this one x is going to equal plus or minus the square root of three for this one we reject the reason why we reject it is because we cannot square root a negative number okay this is not allowed so you would write reject so this graph it would only cross the x axis once okay it would look like this so can you see it only crosses it once but if we worked out it'd have one minimum and one maximum point okay i have already shown you how to sketch a graph and we all seemed fairly confident we need to remember okay these are the roots how do you find the roots and that's by solving the equation okay and we solve the equation by making it equal to zero okay this point here is the y-intercept okay this is normally just the number on its own isn't it at the end okay this is like the plus c this is when x equals zero you'll find the y-intercept you could always substitute it in this is going to be the minimum or the maximum point and how do we get that completing the square okay then this one is just making sure the shape is correct okay if you have a positive x squared you're going to have a u if you have a negative x squared you're going to have a n okay so okay this is nice and sketched this is the one we did earlier okay and we can see we've got our four marks okay here we have a at four x minus two x squared minus three and you can see we've got the sketch here okay we've got minus three one and minus one and the reason why is when we try and do the roots there are no roots it comes up with error we can't factorize it and if we put this into our calculator there is an error so could you try that for me in your calculator your 4x so i'm going to rewrite it as minus set minus 2x squared plus 4x minus 3. so your a equals minus 2 your b equals four and c equals minus three yeah so this should either come up with an error or we're going to get the square root of something and an i which is an imaginary number which you're not going to learn about unless you um do further mathematics or take up a degree in mathematics so don't worry about that but it it doesn't cross the axis if that happens you're not going to have anywhere that crosses along here okay so what i would like you to do now is i want you to make sure you can sketch these graphs there's four of them to sketch it shouldn't take you long to sketch a graph so i am expecting you to do them all okay step one draw your axes and always write x and y step two now i don't need to solve this in any way because x squared graph always looks like this okay and it crosses at zero zero now if you think about a function right do you remember when we were transforming graphs and if we plus four all it does is goes up four because it's not inside the bracket is it inside or outside the bracket sarah outside the bracket so we're just going up four so actually it's going to look like this and this point here is 0 4. and that's it done right then so b we have x squared minus 7 7x plus 10 we can factorize that to get x minus 5 x minus 2 so x equals 5 x equals 2. so those are our roots that's where it crosses the x axis um we can complete the square x minus three uh seven over two squared 49 over four plus is it nine over four yeah so plus nine over four now just to reiterate i've done that in my head okay oh sorry it's minus i've completely done that in my head okay and double checked that it's correct if you do not know how to do this please look at the previous ones i'm completing the square it's just i've missed a step out i've just tried to do it as fast as i can no you don't need to do in your head i'm just doing in my head and then the y insert equals 10. so sorry running down my face all right back again so then so now what we're going to do is we have all the points that we need the coordinates of this one is going to be 7 over 2 and minus 9 over 4. so here we go we have our sketch x and y if you don't have that just mark your homework all is wrong right now because that's not acceptable hazel oh perfect then we know it's going to be two and five so we know it's gonna be about here this is this here is two zero five zero we're going to have 0 10 and then our minimum is going to be 7 over 2 and minus 9 over 4. okay so that is exactly four marks and that is where we should be as a class we should be getting comfortable with that okay that is our minimum expectation and that's gcse and a little extension so well done if you've got that right okay the next one in our sketch um can someone tell me the factorized form please or what am i what are my roots three negative a half okay so if those are my roots i can always put it back in so x is minus three and two x plus one okay i should be confident if we've got the roots we can put that back in this is how we can cheat factorizing okay you know when it factorizes if you can use your calculator to find your roots you can just reverse it and put it back into the equation okay so then um completed form square please minus okay is that right anyone 49 over eight okay so well done sorry minus two times yeah and then yeah you've got a times the 49 over 16 by the two because you're bringing that two back here you multiply everything by two so then again um we have three and minus a half three oh no this doesn't make sense [Music] oh yeah that then it does make sense okay there we go so this is our free zero this here is going to be zero free this point is going to be minus a half and zero don't forget your x and your y and then this point is going to be so this should be positive 49 over right then yeah so this is going to be 5 over 4 and 49 over 8. okay perfect now the last one when we try and factorize what happens error oh it gives you imaginary numbers yeah so that means it's an error okay so there's got no roots that's fine my y intercept is 11. and when i complete the square what do i get x plus 2 squared uh minus 4 plus 11 is plus 7. so we're going to have minus 2 and 7 as our coordinate so when we draw this now because it has no roots okay we can't factorize it it's going to look like -2 minus two seven is roughly here okay so now it's going to look like this so it's crossing here at 11 and this minimum point is -27 it doesn't cross the x-axis so it means it has no roots okay if it doesn't cross the x-axis it doesn't have any roots so that's how the graph is going to look i know some of us didn't get that quite right but at least now we know how to do it thank you so these are the graphs find the equation now this is what i just showed you a little bit of then okay if the roots are free and five you know that x equals 3 and x equals 5 you could just put them back in x minus 3 x minus 5 okay and then if you expand that you're going to get x squared minus 8x plus 15 which makes sense because your y intercepts plus 15. nice and easy it's just reversing okay part b you get x is minus two and minus five so you're gonna get x plus two x minus five okay but if we did two times minus five we're going to notice that we're gonna not get positive ten and if you look at the graph it's a negative so actually it's x plus two and then it's 5 minus x okay or it could be 2 minus x either way one of them is going to be a minus or you could put a minus on the outside okay and what you're going to end up is with minus x squared minus 7 x plus 10. okay so what do we have oh it's yeah you're right sorry yeah okay is that the answer to you right the next one it's three and minus three so you're gonna have x minus three x plus three which is the difference of two squares we're going to get x squared but that doesn't give us minus 18 does it no so why does that not give us minus 18 because this is actually an enlargement so we need to put a 3 sorry a 2 on the outside so we're going to have 2x squared minus 18. okay then for the last one i'm gonna get x minus four x plus one now then that should give me minus four but so we're going to have a quarter of that to get the y-intercept because it's currently it crosses at minus one if i multiply this out i'm gonna get minus four so i need to do a quarter so i'm gonna get x squared over four minus three x over four minus a quarter minus one okay perfect right then so we should all be fairly confident with graphs if not we'll do some more work on that um some of you may have done the extension did anyone do it no i'm gonna go for it then if you want to do it you can okay then so moving on how many distinct real roots now this is new this is new content now everyone everything else we've done so far is what we should know and maybe just a bit uh some of the harder questions if that makes sense okay but this is completely new so how many distinct real solutions do each of the following have can you please either solve them by factorizing or put each one into your calculator to see what the solutions are now when we're looking for solutions and it says real solutions okay so imaginary numbers are not real numbers so we're looking for our solutions i want to know what's going on okay so you can put each one into your calculator you know that your a is always the number in front of the x squared your c is the number in front of the x and uh b is the one in front of the x and you'll see is the other one now i don't like saying the number in front of okay so can we make sure that we understand the term coefficient okay is the number in front of if there's no co if you just have one x squared the coefficient is one okay so i want us to get really used to understanding we also take the sign with the coefficient do not forget about the sign that's probably one of the most biggest mistakes okay so make sure that you are completely happy with this right then so you should have put these into the calculators for the first one what does x equal please six is that it okay for the second one um no real solutions and for the last one one plus or minus the square root of two okay so that means that either you've got x equals 1 plus root 2 or x equals 1 minus root 2. now you've had to sit and work that all out okay but actually we've got a formula that we can use to determine how many real roots an equation has okay so um so um as i said the discriminant is known as b squared minus four ac okay and that's called the discriminant need to learn that so here we go what we have is something called the discriminant okay the discriminant now on the quadratic formula okay we have minus b plus or minus the square root of b squared minus 4ac over 2a this is the quadratic formula okay now the discriminant is this part here oh sorry not including the square root it's this part here okay this is the discriminant right now this there are a few rules okay there's there's only three options okay we can have two real roots oh i've put rots roots okay we can have one root as we've seen but there's another name for this it's called equal roots okay and the reason why it's called equal roots i'll explain in a minute or we can have no real roots okay and that's when we get an imaginary number or we get an error now this happens believe it or not on three different occasions so the discriminant is b squared minus four ac is everyone happy the discriminant is b squared minus 4ac now we get an imaginary number if we try and square root a negative a negative number so for any equation if b squared minus 4ac is less than 0 we get no real roots okay if b squared minus 4ac equals 0 because if this equals 0 here we've got the square root of zero we don't have the plus or minus but we're just going to get minus b over 2a so if it equals zero we're only going to get one root or equal roots okay and then lastly then if b squared minus four ac is greater than zero we're going to have two real roots so how what this actually means i'm talking about all these roots and everything going on what this means is when i have my graph if i have two real roots it's going to cross the x-axis at two points okay if i have one root or equal root it's going to cross the x-axis at one point okay and if i have no equal roots what's going to happen is it's not going to cross the axes at all and all i'm talking about is the x axis here it's really simple so again here we've got um just a few more pictures to show you that as long as you understand exactly what each one means and what it represents on a graph so and the reason why we say equal roots are not one root you can um actually read through this at a later date but just very quickly if i was to factorize this one i'm gonna get x minus six x minus six equals zero so i actually have two lots of x minus six don't i so that in theory you know that's a kind of short of it it's equal roots because six and six they're equal to each other but really it's just one root okay but have a look a bit more into that for those who want to um challenge yourselves so then what we're going to do very quickly is we're going to do some quick fire questions okay the discriminant is b squared minus 4ac so for the first one to calculate it and i'm just going to make sure everyone's happy remember your a b c okay so for the first one i'm going to have b squared which is now i'm always and this is very important whenever i substitute i use brackets 3 squared minus 4 bracket 1 bracket full okay now some of you might be able to do the discriminant in your head for each one but i just want to make sure that we are completely happy with how this looks so you know where every point has come from okay and then if we do this we're going to get 9 minus 16 okay 9-16 is nine minus sixteen thank you minus seven okay this is less than zero if it's less than zero we have no roots because it's less than zero okay you've got one two three four five to do off you go okay then so b squared minus four ac so i'm gonna do this in my head so i'm gonna get 16 minus four that's positive so i'm going to have two roots okay i don't actually because these are just quick questions i don't actually need to work out the entire number it's good that you know how to and if you are so we are going to do 12 is greater than zero okay which is two roots everyone happy okay um b squared so 16. minus 16 i'm gonna get equals zero that means it has one root or equal roots now i'm going to get 36 plus this is going to be greater than zero isn't it two roots okay and that's because i'm doing six squared which is thirty negative six squared which minus 4 lots of a which is 2 minus 3. i know that i've got a positive plus a positive okay i don't even have to work that out okay for the next one again i'm going to have one squared minus four lots of minus three lots of minus four that's gonna give me a negative number isn't it it's less than zero so it's got um no roots then our b is zero so we're going to zero squared minus four lots of one minus one lots of one which is positive so two roots okay because if you think don't forget everybody your a is pink okay your b is green well we don't have a b in this case it's zero um and your c is the coefficient make sure you understand exactly what's not the coefficient a integer okay make sure you understand exactly what's going on so here we have um an exam style question this is six marks it's such a lovely question for six marks okay really lovely so i'm gonna do this question you don't have to write it you're gonna have your own questions to do in a minute i just want everyone focusing and listening again you can take pictures that you know pictures and whatever of this can be done later this is all saved as a video so are we ready here we go the equation x squared plus two p x plus three p plus four equals zero where p is a positive constant has equal roots in the question if it ever states something has equal roots the first thing that you should come to mind is that b squared minus four ac equals zero that is the first thing that should come to mind equal roots b squared minus four ac equals zero that's i want you to live and breathe that okay that's what i want so again we're going to use the color coding your a is the coefficient in front of the x squared which in this case is 1. so you know i would all you need to write this down i would always suggest so your a is 1 perfect your b is the coefficient in front of the x which in this case is positive 2p this is oh 2 p or to be or not to be i don't know right then the coefficient at the c is our constant okay it's the one that doesn't have an x near it it's just a number and in this case it's this okay so everyone should be happy and again use some highlighters if you're getting confused what's going on three p plus four 2 p okay so then it says find the value of p well it's really easy to find p because you know that b squared minus 4 ac equals 0. so we're going to have b squared now i've i keep saying to you make sure we are writing brackets this is b squared minus 4ac so our b is 2p our a is 1 and our c is 3p plus 4 and we've got to find p so now 2p squared is 4p squared minus 4 lots of 3p plus 4 because i multiplied it by 1 okay which is going to give me 4p squared minus and remember this equals zero because we've got equal roots doesn't it uh minus 12 p minus 16 equals zero okay believe it or not what we got a quadratic i can't even believe it can you right so what we're going to do is we're going to solve it um p squared minus 3. so i'm going to get p minus four p plus three equals zero uh not three sorry one equals zero i was thinking it has to give me free because i divided everything by four again i'm just doing this in my head you can do this on a calculator so our p equals uh four or p equals minus one nice and easy four marks right there four marks for doing that and i expect you to show all of these steps okay okay so that was a good question then the jeep asked what would happen if a was for example minus two here it's lucky it's one so i've just got minus four lots of one which is why i've got minus four here if it was minus two i'd get minus four lots of minus two which is plus eight and then i'd have eight lots of the brackets okay so i've got p equals four p equals minus one so for this value of p solve the equation x squared plus two p x plus three p plus four equals zero we have two values for p so this is just imagine it's like quadratic um simultaneous equations okay we now have two values of p i've got to put those two values of p into the original equation to get two outputs so um if we let p equal four then our equation is going to be x squared plus two lots of 4x because instead of p we've said it's 4 plus 3 lots of p plus 4 equals 0. so that's going to give me x squared plus 8x 12 plus 16 equals zero i'm going to factorize this solve this i'm going to get x plus four squared so x equals minus four okay now again i have done that in my head you need to get to that stage by the end of december you will but use your calculator now if p equals minus one we're gonna have x squared plus two lots of minus one x plus three lots of minus one plus four so that's actually going to give me x squared minus two x three times minus one is minus three plus four which is minus one which equals zero but i think that also gives us x minus one no no i don't think we can solve this yeah this is plus oh this is a plus one oh there we go so x minus one x minus one so this is x minus one squared equals zero so x equals one so we have two solutions x equals minus four and x equals one so what i would like you to do now is you have two questions to do test your understanding everyone start with the green and then move on to the yellow okay then i'm just going to quickly go through these um and then after that um we finished chapter one okay we've got a few questions to do great we finished chapter two then even better right it has equal roots what's the first thing we should be writing down that b squared minus 4ac equals zero okay now our b is whatever is in front of the x and the coefficient of x so we're going to have k squared minus 4 lots of a which is 1 lots of c which is 10 k plus 5 equals 0. so i'm going to get 25 k squared minus 40k minus 20 equals zero and if i factorize that what do i get x equals two and x equals minus two over five okay but it's not x is it it's k nice and simple that's the first one done and again that's worth full marks four marks for that really really nice and easy right the next one's slightly different and this is why it's covered in orange because some of you may or may not know the answer to this find the range of values of k for which x squared plus six x plus k equals zero has two distinct solutions it has two solutions b squared minus four ac is greater than zero we've just written down those rules okay my b is in fact six squared minus four lots of a which is one lots of c which in this case is k that is greater than zero okay so i'm going to get 36 minus 4k is greater than zero now when you solve this we can do this a few different ways okay i want k on its own right so we could um move the 36 on to the other side so we're going to get minus 4k is greater than minus 36. if we divide by minus 4 though what do we have to do change the sign well done so then if i divide both sides by minus 4 i get k is less than nine so what this is saying then is for this to have two distinct solutions k has to be less than nine okay so let's actually give this just a bit of a go okay so this is just for you know being curious mathematicians as we are let's try k equals 10 x squared plus 6x plus 10 okay what happens if k equals 10. if you put that into your calculator it should give you an imaginary number ever shouldn't it okay so this is going to give you error okay no solutions no roots okay if we put it equals nine we should find that we have um equal roots so if we let k equal nine what's going to happen yeah we put that into our calculator we should find that the root is free shouldn't we that's going to give us equal roots and then if we try 8 if we try 8 plus eight we're going to get two roots okay so this is showing us that for any value this tells us that k has to be less than nine in this specific equation so that we have two real solutions okay okay if you're feeling good feeling really good you need to do five six seven if you don't finish it in the lesson you finish it for homework okay the most important thing is five six seven if you're not feeling confident do two three four but really everyone should be feeling super confident um there are some challenge questions for house points we don't have to worry about them and there is a further challenge for this whole chapter again house points