Understanding Discriminants in Quadratic Equations

Okay, so in this video we're going to have a look at everything to do with the discriminants. Now the discriminants is something that you have to deal with. met before, 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 in 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 but obviously 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 4ac and that is what we call a discriminant. call the discriminant. So we're just going to be having a look at this bit. 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 we can have three different types of numbers. We can have a whole number and let's pick something that actually... square root something like 25 and the square root of 25 would give us two solutions we can get plus 5 and we can get minus 5. 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 0, and if we get the number 0, obviously the square root of 0 is 0, 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 0, then both of our solutions are going to 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, we'll 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 4x plus 2. 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 you're 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've got a value of b squared minus 4ac, so we 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 8. So the value of our discriminant is 8. And if we think about that, if it was put into the quadratic formula, we would have plus and minus root 8 which would give us our two values underneath the square root there. that would give us our two solutions and ultimately determine these two solutions. So in terms of stating whether this, 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. But let's have a look at one that's slightly different. Okay, so for this one here, you can see already on the graph that we only have one root and it touches here on the graph you can see it's at 3 and when we've got something like this we say there's two equal real roots okay and that's because both of the values are the same okay but technically looking at the graph we do only have the one roots or the one solution but we refer to that as having two equal real roots okay because when we obviously do that with the plus and the minus they just both end up being the same but if we have a look at applying this again our value of a is 1 again so a is 1 our value of B here is negative negative 6 and our value of c is positive 9. And if we put that into our b squared minus 4ac and we'll see what we get. Now let's have a look. So b squared is negative 6 squared, so obviously just being careful when you square a negative, get that in a bracket. And we're going to take away 4 times 1 times 9. There we go. So negative 6 squared is 36, so we've got 36 for that. And we're going to take away 4 times 1 times 9. which is 36. So 36, take away 36 is 0. There we go. And as you can see, obviously we discussed it at the start, if we have the square root of 0 underneath our square root there, we're going to get two answers that are the same when using the quadratic formula. So there we go, we get two equal roots. There we are, so two equal Roots, okay, and obviously thinking about it logically in terms of the graph there, obviously our root there is at 3. So there we go, that is how we would determine obviously the value of the discriminant for this one, and then stating whether that would have obviously 2. roots, two equal roots or no real roots. And we're going to have a look at our third one here which is going to have no real roots. So for this question here then you can obviously see on the graph we've got no roots. Our graph is up here, way above the x-axis. and we're just going to show that obviously using a discriminant how we can show that it has no real roots without obviously using the graph. So if we get our values of a, b and c again, this time we do have a value of a greater than 1. We've got a is 2, b is negative 1, the number in front of x there, minus x is minus 1 and c equals 3. So if we put those in again, we've got b squared minus 4ac and we're going to put these values into that again. So negative 1 squared for b, we have minus 1 squared and we're going to... to take away 4 times a which is 2 times c which is 3. So negative 1 squared is positive 1 and we're going to take away 4 times 2 times 3. 4 times 2 is 8, times 3 is 24 so it's 1 take away 24. Right, so one take away 24 obviously comes out as negative 23. And there we go, thinking about that in terms of underneath the square root, if we try and square root a negative, there is no answer for that one there, so we don't get an answer, so we would have no real roots for this one, which obviously we can see from the graph, but we have no real roots. Okay, there we go. So there are three examples and hopefully a little bit of understanding as to why the positive number comes out as two roots, why a zero comes out as having two equal roots or the one root and why a negative number comes out as having no real roots. Right. Okay. So here's a couple of questions for you to have a go at before we move on. Okay, so there's two questions here. So can you find the value of the discriminant for both of them and then also state whether they have two distinct roots, equal roots or no roots? real roots. So pause the video there, have a go and we'll go over the answer in a sec. So for the first one then, finding the value of the discriminant here, and not forget to always write down b squared minus 4ac. And our value of a for this one is 4, our value of b is negative 4, and our value of c is 1. And if we stick those all into the discriminant there we get negative 4 squared. Take away 4 times a which is 4 times c which is 1. So negative 4 squared is 16. Take away 4 times 4 times 1 is also 16. So that equals 0. There we go. So we have two equal roots. There we go, two equal roots for our first one. And let's just highlight that so we get 0 and two equal roots. Onto our next one, let's have a look. So we've got b squared minus 4ac again. Our value of a is 3, our value of b is negative 2, and our value of c is negative 5. And if we put these in, let's see what we get. So negative 2 squared take away 4 times a which is 3, times c which is negative 5. Got to be a little bit careful here with that negative 5 at the end. So negative 2 squared is 4. And then we are going to take away, and I'm just going to swap colours for this one, we've got 4 times 3 times negative 5. Now 4 times 3 is 12, times negative 5 is negative 60. So we've got 4 take away negative 60. And that's OK because that turns into a plus, so we have 4 plus 60, which gives us the answer 64. There we go. And that is obviously going to give us two real roots, or two roots. There we go, two distinct roots. There we go, two distinct roots and that is going to be our final answer for that one. Okay, so there we go, 64 was our discriminant and that gave us two distinct roots. So that's how we're going to look at quadratics and that's how we're going to determine how many roots they have and whether they have a root at all. So let's have a look at a slightly harder question than looking at this topic of the discriminant. Okay, so this question says the equation px squared plus 2x minus 3 equals 0. where p is a constant has equal roots find the value of p. Now we know when something has equal roots it equals zero. So, well the discriminant equals zero sorry, so if we know that the discriminant equals zero we can actually extract some of these pieces from the quadratic and try and figure out that value of p because if we're told it has equal roots we know that b squared minus 4ac has to equal zero. So if we try and get these pieces from here then and see what we have. So we've got a at the moment which is a letter, we've got a equals p, we've got b equals positive 2 and c equals negative 3. So essentially we just want to find out what value of p is going to ensure that b squared minus 4ac definitely equals 0. So if we put these pieces in then we get 2 squared, take away, I'll keep that in a bracket, take away 4 times p times negative 3. and that is going to equal zero. Okay so let's tidy this up, 2 squared is 4 and we're going to take away and this here is going to equal 4 times 3, 4 times negative 3 is negative 12 so we have minus 12p there. So we're going to take away negative 12p which is going to become plus 12p, there we go so we have 4 plus 12p that has to equal zero and now we've just got a nice simple equation to solve so we can take away 4 from both sides we get 12p equals negative 4, divide both sides by 12, so p equals negative 4 over 12, and that simplifies down to minus a third, and there is our value of p negative a third. So there we go, that's how we're going to go about obviously approaching some of these questions here where the wording is slightly different. It obviously doesn't say in the question anything to do with the discriminant, but obviously we're using that knowledge of knowing that it has equal roots and applying the discriminant towards that. So let's have a look at one more of these before you have a go. So this question is slightly trickier than the way that it looks, but we're going to take a very similar approach. So it says here that this equation, where q is a constant, has equal roots, find the value of q. So again, if it has equal roots, we know that b squared minus 4ac has to equal 0. And if we extract the pieces from here again, we've got a this time equals 3q, what's in front of x squared. We've got b is equal to negative 4q. And we have c is equal to positive 4. So not very nice at all in the way that it looks. But let's just have a look at going about this. So b squared. So we have negative 4q is our b, and that's going to be squared. And we are going to take away 4 times 3q times another 4. There we go. And that is all going to equal 0. Now let's expand this all out. So negative 4q squared. will become positive 16q squared. And we're going to take away, and we just need to multiply all of this. So 4 times 3 is 12, times 4 is 48. So we have 48q there. There we go. So there's obviously no more negatives in there, so let's just take away 48q. There we go, and that all equals 0. Now we can go about solving this in different ways, but essentially we can actually factorise this what's on the left here and then divide, or we could divide both sides by some factors. We could divide both sides by q to start with if you want to take it in steps, so if we divide both sides by q that would be 16q minus 48 equals 0. It's not the nicest to spot but they obviously do both divide by 16. Okay obviously you could solve this a little bit slower and not spot that but they do actually both divide by 16. So we could divide both sides by 16 now. So we'd get Q minus 3 equals 0. And then obviously we can add that 3 over to the other side and we get q equals 3. There we go. And there is our final answer for that one as well. So sometimes you might get fractions, sometimes you might get whole numbers, or you could even get negatives. But obviously just watching out for solving that equation there and just being nice and careful when you do so. But there we go, that is obviously some of these slightly harder questions. We are going to have a little go at this now before moving on to our last style of question, but here's a couple for you to have a go at. Okay so there's two questions here, so pause the video there, have a go and we'll go over the answers in a sec. Okay so for this first one then. So we've got b squared minus 4ac. and our value of a is p, b is 4 and c is negative 2. And if we put those in we get 4 squared minus 4 times p times negative 2. And then again, simplifying that down gives us 16, and we're going to take away minus 8p. Okay, so take away minus 8p would be plus 8p. So we have 16 plus 8p, and that all equals 0. Obviously because it says it has equal roots we can put that equal 0 up there as well if we want. So taking away 16 from both sides we get 8p equals negative 16. And then dividing both sides by 8 gives us p equal to negative 2. And there we go and there's our final answer for that first one. On to the next one. Again we've got equal roots so b squared minus 4ac and that is going to equal 0. And we're going to have a look obviously just extracted the... pieces again even though this one looks a little bit harder. So let's have a look. We've got a equals 1, we've got b equals 4q and we've got c equals 2q. Right okay so let's put that in. We've got 4q squared for b so we've got 4q squared. squared minus 4 times 1 times 2q. Right, there we go. So let's obviously just tidy that up a bit. 4q squared is going to equal 16q squared. And then this bit here is going to equal, let's have a look, we're taking that away. 4 times 2 times 1 is 8, so take away 8q. There we go, so 16q squared, take away 8q, is equal to 0. So, there we go, we can divide both sides. sides by q we could divide both sides by 8q if we wanted but let's just take it in steps I'll divide both sides by q to start with so that'll give me 16q minus 8 equals 0 divide both sides by 8 we get 2q equal sorry 2q I was getting ahead of myself there minus 1 equals 0 add the 1 to the other side 2q equals 1 and then divide both sides by 2 we get q equals a half There we go and there's our final answer for that one. Obviously you could actually just move that 8 over or move the 8q over right at the start and solve it in a slightly different way. It's completely up to you what steps you use in terms of your solving equations there but as long as you get to that final answer of a half it's absolutely fine to do your own method. But there we go that is that bit let's have a look at something different. Okay so I'm only going to go over one of these questions there's a lot of different elements or different parts to this one and then you're going to have a go at one before we finish. So this one says here here the function of x is x squared plus, and then our coefficient of x is k plus 5, and our c value at the end there is 2k, where k is a constant. So let's find the discriminant of f in terms of k. So before we move on, let's have a look at that bit. So finding the discriminant for that's going to be okay, because all we have to do is b squared minus 4ac. And as we can see from up here, a equals 1, b equals k plus 5, and then we which I'll leave in a bracket, and c equals 2k. Right, there we go. So, plugging these pieces into b squared minus 4ac then, b is k plus 5, so we're going to have k plus 5... squared minus 4 times a times c which is 4 times 1 which is 4 times 2k. So 4 times 2k is 8k. There we go. So we have k plus 5 in brackets squared minus 8k and that is the discriminant of the function of x in terms of k. Obviously we've just got our k's in there, we don't need to expand that out or tidy it up, that is my discriminant and I can just leave it as it is. Now let's just underline that, that's because I'll put First answer, and let's have a look at part B here. So part B says show that the discriminant can be written in the form k plus p squared plus q. Now you'll hopefully recognise that that is in completed square form. So essentially what we're going to do is complete the square. So obviously I will link the video below for completing the square because obviously you need to know Obviously that topic before we move on and have a look at this question So if you're not sure on completed square form or you don't recognize that Do make sure you go check out the video and then maybe come back to this because I'm going to assume that you know know how to complete the square for this. So it says where p and q are integers. Now obviously we've got our discriminant up here, so if we go about expanding that, we'll see what we get. So at the moment we've got k plus 5 in brackets squared minus 8k. And if I expand that out, let's have a look, that k plus 5 squared is a double bracket, and we get k squared plus 2 lots of 5k, so plus 10k, plus 25. and then we're going to minus that 8k at the end. So if we tidy all that up, look, we've got these two bits here that we can simplify, the 10k minus 8k. So if we simplify that down, we get k squared plus 2k plus 25. So that's when we expand it all out and tidy it up. Now obviously now we need to complete the square. So we can complete the square, and it's not a particularly nasty one to complete the square for. If we half the coefficient of k there, we get k plus 1k. one in brackets squared that would give us one when we expand it we want 25 so we need an extra 24 there we go and that is that in completed square form so p is equal to one and q is equal to 24 there we go so that is in completed square form and that is our answer to part two and then for the next bit here we're going to have a look at it says show that four and let's just swap colors now so show that for all values of k the equation where the the function of x equals zero has distinct real roots. So when it comes to this, essentially all we're having to look at is this discriminant that we've got written in completed square form. If we write that out, we've got k plus 1 squared plus 24, and we've got to show that that has distinct real roots. So when that equation equals zero, we've got the discriminant here that we're looking at, and essentially all we're looking at is the piece here within the bracket. Because if we think about this, this number in here, if I was to... sub something in, and bearing in mind what distinct real roots means, it means that we've got to have a number greater than zero for our discriminant. If you think about substituting a number into here, it doesn't matter what number I put into there, that is always going to be bigger than or equal to zero. to 0 or greater than or equal to 0. If we put negative 1 in there that would give us the answer 0 but anything less than negative 1 would give us a negative number in the bracket which when squared would be positive and any positive number in there would also give us a positive number as well when we square it. So if we think about that even if it was negative 1 in there we would then add the 24 and that would make that number positive. So because this is always going to be positive, it's always going to be bigger than zero because of that plus 24 there, we can make the statement that b squared minus 4ac, the discriminant, is always going to be greater than zero and therefore, using our little therefore symbol, it always has distinct real roots. Always has distinct real roots. There we go and we can just write that out. Okay, and that is literally all we need to do to prove that there. Just obviously making that statement that obviously that discriminant there is always going to be bigger than 0, and therefore it's always going to have distinct real roots just using that completed square form that we've done on the previous question. So there we go, that's how we go about answering these three elements of this particular question here. We had a look at obviously showing the discriminant using these values of k that we had in our quadratic up there. We then obviously looked. at writing it in completed square form, so expanding and simplifying it and completing the square, and then obviously using that to show that it's always going to be bigger than zero. So that's how we're going to go about answering a question like this, and now here's one for you to have a go at. Okay, so here's your question, so pause the video there, have a go, and we'll go over the answer in a sec. Okay, so for the first thing here, we're going to find the discriminant here. So we have a equals 1, b is equal to k plus 3, and c is equal to that k at the end there. So if we put this in terms of k for the discriminant, remember when we're doing b squared minus 4ac. There we go. So that's k plus 3 for b, so we have k plus 3. squared, there we go, and minus 4 times 1 times k. So that's going to be minus 4k. There we go. And that is my discriminant in terms of k. OK, there we are, let's just highlight that. Right, so for the next part, it says, show that the discriminant can be written in this completed square form where p and q are integers. There we go, so let's expand that out and complete the square. So when we expand it out, and let's just write this down here. so we have k plus 3 squared minus 4k and if we expand that double bracket there we get k squared plus 2 lots of 3k so plus 6k and then 3 times 3 is 9 so plus 9 and then we're also going to minus this 4k at the end. So again let's simplify that and collect these two terms here and we will get k squared 6 take away 4 is 2 so plus 2k and then plus the 9. Now again we just need to complete the square for that so halving the coefficient of k and we get k plus 1 squared that gives us 1 we want 9 so plus 8 at the end. There we are so that's it written in completed square. form. P is 1 and Q is 8. There we are, so finishing this off, it now says, obviously similar to last time, show that for all values of k, the equation where f of x equals 0 has real roots. So again, looking at our completed square form here, at the moment we have k plus 1 squared plus 8, and again we're looking at that part in the bracket there. So this part in the bracket, again, is always going to have a value, I'm just going to write always bigger than or equal to 0. equal to 0 and obviously because we've got the plus 8 there even if it was negative 1 subbed into that our value of the discriminant which is b squared minus 4ac is always going to be greater than 0 and because it's always greater than 0 we can say therefore always has real roots There we go, and we can finish that off just by writing that, and there we go, we're done. Okay, so that is obviously all the work on the discriminant that we're going to have a look at. Obviously we looked at lots of different bits of maths there, so as I said before, I will put the link in the description for work on the quadratic formula. and I'll also put a couple of videos linking to obviously completing the square and some of the harder completing the square types of questions that you can have as well obviously just make sure that you do check those out make sure that you've really clued up on all those topics but there we go that's the end of the discriminant hopefully that was useful and helpful if it was please like please comment please subscribe and I'll see you for the next one

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 in 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 but obviously 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 4ac and that is what we call a discriminant.

call the discriminant. So we're just going to be having a look at this bit. 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 we can have three different types of numbers.

We can have a whole number and let's pick something that actually... square root something like 25 and the square root of 25 would give us two solutions we can get plus 5 and we can get minus 5. 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 0, and if we get the number 0, obviously the square root of 0 is 0, 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 0, then both of our solutions are going to 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, we'll 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 4x plus 2. 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 you're 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've got a value of b squared minus 4ac, so we 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 8. So the value of our discriminant is 8. And if we think about that, if it was put into the quadratic formula, we would have plus and minus root 8 which would give us our two values underneath the square root there. that would give us our two solutions and ultimately determine these two solutions. So in terms of stating whether this, 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. But let's have a look at one that's slightly different. Okay, so for this one here, you can see already on the graph that we only have one root and it touches here on the graph you can see it's at 3 and when we've got something like this we say there's two equal real roots okay and that's because both of the values are the same okay but technically looking at the graph we do only have the one roots or the one solution but we refer to that as having two equal real roots okay because when we obviously do that with the plus and the minus they just both end up being the same but if we have a look at applying this again our value of a is 1 again so a is 1 our value of B here is negative negative 6 and our value of c is positive 9. And if we put that into our b squared minus 4ac and we'll see what we get.

Now let's have a look. So b squared is negative 6 squared, so obviously just being careful when you square a negative, get that in a bracket. And we're going to take away 4 times 1 times 9. There we go. So negative 6 squared is 36, so we've got 36 for that. And we're going to take away 4 times 1 times 9. which is 36. So 36, take away 36 is 0. There we go.

And as you can see, obviously we discussed it at the start, if we have the square root of 0 underneath our square root there, we're going to get two answers that are the same when using the quadratic formula. So there we go, we get two equal roots. There we are, so two equal Roots, okay, and obviously thinking about it logically in terms of the graph there, obviously our root there is at 3. So there we go, that is how we would determine obviously the value of the discriminant for this one, and then stating whether that would have obviously 2. roots, two equal roots or no real roots. And we're going to have a look at our third one here which is going to have no real roots.

So for this question here then you can obviously see on the graph we've got no roots. Our graph is up here, way above the x-axis. and we're just going to show that obviously using a discriminant how we can show that it has no real roots without obviously using the graph. So if we get our values of a, b and c again, this time we do have a value of a greater than 1. We've got a is 2, b is negative 1, the number in front of x there, minus x is minus 1 and c equals 3. So if we put those in again, we've got b squared minus 4ac and we're going to put these values into that again. So negative 1 squared for b, we have minus 1 squared and we're going to...

to take away 4 times a which is 2 times c which is 3. So negative 1 squared is positive 1 and we're going to take away 4 times 2 times 3. 4 times 2 is 8, times 3 is 24 so it's 1 take away 24. Right, so one take away 24 obviously comes out as negative 23. And there we go, thinking about that in terms of underneath the square root, if we try and square root a negative, there is no answer for that one there, so we don't get an answer, so we would have no real roots for this one, which obviously we can see from the graph, but we have no real roots. Okay, there we go. So there are three examples and hopefully a little bit of understanding as to why the positive number comes out as two roots, why a zero comes out as having two equal roots or the one root and why a negative number comes out as having no real roots.

Right. Okay. So here's a couple of questions for you to have a go at before we move on.

Okay, so there's two questions here. So can you find the value of the discriminant for both of them and then also state whether they have two distinct roots, equal roots or no roots? real roots. So pause the video there, have a go and we'll go over the answer in a sec.

So for the first one then, finding the value of the discriminant here, and not forget to always write down b squared minus 4ac. And our value of a for this one is 4, our value of b is negative 4, and our value of c is 1. And if we stick those all into the discriminant there we get negative 4 squared. Take away 4 times a which is 4 times c which is 1. So negative 4 squared is 16. Take away 4 times 4 times 1 is also 16. So that equals 0. There we go.

So we have two equal roots. There we go, two equal roots for our first one. And let's just highlight that so we get 0 and two equal roots.

Onto our next one, let's have a look. So we've got b squared minus 4ac again. Our value of a is 3, our value of b is negative 2, and our value of c is negative 5. And if we put these in, let's see what we get. So negative 2 squared take away 4 times a which is 3, times c which is negative 5. Got to be a little bit careful here with that negative 5 at the end.

So negative 2 squared is 4. And then we are going to take away, and I'm just going to swap colours for this one, we've got 4 times 3 times negative 5. Now 4 times 3 is 12, times negative 5 is negative 60. So we've got 4 take away negative 60. And that's OK because that turns into a plus, so we have 4 plus 60, which gives us the answer 64. There we go. And that is obviously going to give us two real roots, or two roots. There we go, two distinct roots.

There we go, two distinct roots and that is going to be our final answer for that one. Okay, so there we go, 64 was our discriminant and that gave us two distinct roots. So that's how we're going to look at quadratics and that's how we're going to determine how many roots they have and whether they have a root at all. So let's have a look at a slightly harder question than looking at this topic of the discriminant.

Okay, so this question says the equation px squared plus 2x minus 3 equals 0. where p is a constant has equal roots find the value of p. Now we know when something has equal roots it equals zero. So, well the discriminant equals zero sorry, so if we know that the discriminant equals zero we can actually extract some of these pieces from the quadratic and try and figure out that value of p because if we're told it has equal roots we know that b squared minus 4ac has to equal zero.

So if we try and get these pieces from here then and see what we have. So we've got a at the moment which is a letter, we've got a equals p, we've got b equals positive 2 and c equals negative 3. So essentially we just want to find out what value of p is going to ensure that b squared minus 4ac definitely equals 0. So if we put these pieces in then we get 2 squared, take away, I'll keep that in a bracket, take away 4 times p times negative 3. and that is going to equal zero. Okay so let's tidy this up, 2 squared is 4 and we're going to take away and this here is going to equal 4 times 3, 4 times negative 3 is negative 12 so we have minus 12p there. So we're going to take away negative 12p which is going to become plus 12p, there we go so we have 4 plus 12p that has to equal zero and now we've just got a nice simple equation to solve so we can take away 4 from both sides we get 12p equals negative 4, divide both sides by 12, so p equals negative 4 over 12, and that simplifies down to minus a third, and there is our value of p negative a third.

So there we go, that's how we're going to go about obviously approaching some of these questions here where the wording is slightly different. It obviously doesn't say in the question anything to do with the discriminant, but obviously we're using that knowledge of knowing that it has equal roots and applying the discriminant towards that. So let's have a look at one more of these before you have a go.

So this question is slightly trickier than the way that it looks, but we're going to take a very similar approach. So it says here that this equation, where q is a constant, has equal roots, find the value of q. So again, if it has equal roots, we know that b squared minus 4ac has to equal 0. And if we extract the pieces from here again, we've got a this time equals 3q, what's in front of x squared. We've got b is equal to negative 4q. And we have c is equal to positive 4. So not very nice at all in the way that it looks.

But let's just have a look at going about this. So b squared. So we have negative 4q is our b, and that's going to be squared.

And we are going to take away 4 times 3q times another 4. There we go. And that is all going to equal 0. Now let's expand this all out. So negative 4q squared. will become positive 16q squared.

And we're going to take away, and we just need to multiply all of this. So 4 times 3 is 12, times 4 is 48. So we have 48q there. There we go.

So there's obviously no more negatives in there, so let's just take away 48q. There we go, and that all equals 0. Now we can go about solving this in different ways, but essentially we can actually factorise this what's on the left here and then divide, or we could divide both sides by some factors. We could divide both sides by q to start with if you want to take it in steps, so if we divide both sides by q that would be 16q minus 48 equals 0. It's not the nicest to spot but they obviously do both divide by 16. Okay obviously you could solve this a little bit slower and not spot that but they do actually both divide by 16. So we could divide both sides by 16 now. So we'd get Q minus 3 equals 0. And then obviously we can add that 3 over to the other side and we get q equals 3. There we go.

And there is our final answer for that one as well. So sometimes you might get fractions, sometimes you might get whole numbers, or you could even get negatives. But obviously just watching out for solving that equation there and just being nice and careful when you do so. But there we go, that is obviously some of these slightly harder questions.

We are going to have a little go at this now before moving on to our last style of question, but here's a couple for you to have a go at. Okay so there's two questions here, so pause the video there, have a go and we'll go over the answers in a sec. Okay so for this first one then. So we've got b squared minus 4ac.

and our value of a is p, b is 4 and c is negative 2. And if we put those in we get 4 squared minus 4 times p times negative 2. And then again, simplifying that down gives us 16, and we're going to take away minus 8p. Okay, so take away minus 8p would be plus 8p. So we have 16 plus 8p, and that all equals 0. Obviously because it says it has equal roots we can put that equal 0 up there as well if we want.

So taking away 16 from both sides we get 8p equals negative 16. And then dividing both sides by 8 gives us p equal to negative 2. And there we go and there's our final answer for that first one. On to the next one. Again we've got equal roots so b squared minus 4ac and that is going to equal 0. And we're going to have a look obviously just extracted the...

pieces again even though this one looks a little bit harder. So let's have a look. We've got a equals 1, we've got b equals 4q and we've got c equals 2q.

Right okay so let's put that in. We've got 4q squared for b so we've got 4q squared. squared minus 4 times 1 times 2q. Right, there we go. So let's obviously just tidy that up a bit.

4q squared is going to equal 16q squared. And then this bit here is going to equal, let's have a look, we're taking that away. 4 times 2 times 1 is 8, so take away 8q. There we go, so 16q squared, take away 8q, is equal to 0. So, there we go, we can divide both sides.

sides by q we could divide both sides by 8q if we wanted but let's just take it in steps I'll divide both sides by q to start with so that'll give me 16q minus 8 equals 0 divide both sides by 8 we get 2q equal sorry 2q I was getting ahead of myself there minus 1 equals 0 add the 1 to the other side 2q equals 1 and then divide both sides by 2 we get q equals a half There we go and there's our final answer for that one. Obviously you could actually just move that 8 over or move the 8q over right at the start and solve it in a slightly different way. It's completely up to you what steps you use in terms of your solving equations there but as long as you get to that final answer of a half it's absolutely fine to do your own method. But there we go that is that bit let's have a look at something different.

Okay so I'm only going to go over one of these questions there's a lot of different elements or different parts to this one and then you're going to have a go at one before we finish. So this one says here here the function of x is x squared plus, and then our coefficient of x is k plus 5, and our c value at the end there is 2k, where k is a constant. So let's find the discriminant of f in terms of k.

So before we move on, let's have a look at that bit. So finding the discriminant for that's going to be okay, because all we have to do is b squared minus 4ac. And as we can see from up here, a equals 1, b equals k plus 5, and then we which I'll leave in a bracket, and c equals 2k.

Right, there we go. So, plugging these pieces into b squared minus 4ac then, b is k plus 5, so we're going to have k plus 5... squared minus 4 times a times c which is 4 times 1 which is 4 times 2k. So 4 times 2k is 8k. There we go.

So we have k plus 5 in brackets squared minus 8k and that is the discriminant of the function of x in terms of k. Obviously we've just got our k's in there, we don't need to expand that out or tidy it up, that is my discriminant and I can just leave it as it is. Now let's just underline that, that's because I'll put First answer, and let's have a look at part B here.

So part B says show that the discriminant can be written in the form k plus p squared plus q. Now you'll hopefully recognise that that is in completed square form. So essentially what we're going to do is complete the square.

So obviously I will link the video below for completing the square because obviously you need to know Obviously that topic before we move on and have a look at this question So if you're not sure on completed square form or you don't recognize that Do make sure you go check out the video and then maybe come back to this because I'm going to assume that you know know how to complete the square for this. So it says where p and q are integers. Now obviously we've got our discriminant up here, so if we go about expanding that, we'll see what we get. So at the moment we've got k plus 5 in brackets squared minus 8k. And if I expand that out, let's have a look, that k plus 5 squared is a double bracket, and we get k squared plus 2 lots of 5k, so plus 10k, plus 25. and then we're going to minus that 8k at the end.

So if we tidy all that up, look, we've got these two bits here that we can simplify, the 10k minus 8k. So if we simplify that down, we get k squared plus 2k plus 25. So that's when we expand it all out and tidy it up. Now obviously now we need to complete the square.

So we can complete the square, and it's not a particularly nasty one to complete the square for. If we half the coefficient of k there, we get k plus 1k. one in brackets squared that would give us one when we expand it we want 25 so we need an extra 24 there we go and that is that in completed square form so p is equal to one and q is equal to 24 there we go so that is in completed square form and that is our answer to part two and then for the next bit here we're going to have a look at it says show that four and let's just swap colors now so show that for all values of k the equation where the the function of x equals zero has distinct real roots.

So when it comes to this, essentially all we're having to look at is this discriminant that we've got written in completed square form. If we write that out, we've got k plus 1 squared plus 24, and we've got to show that that has distinct real roots. So when that equation equals zero, we've got the discriminant here that we're looking at, and essentially all we're looking at is the piece here within the bracket.

Because if we think about this, this number in here, if I was to... sub something in, and bearing in mind what distinct real roots means, it means that we've got to have a number greater than zero for our discriminant. If you think about substituting a number into here, it doesn't matter what number I put into there, that is always going to be bigger than or equal to zero.

to 0 or greater than or equal to 0. If we put negative 1 in there that would give us the answer 0 but anything less than negative 1 would give us a negative number in the bracket which when squared would be positive and any positive number in there would also give us a positive number as well when we square it. So if we think about that even if it was negative 1 in there we would then add the 24 and that would make that number positive. So because this is always going to be positive, it's always going to be bigger than zero because of that plus 24 there, we can make the statement that b squared minus 4ac, the discriminant, is always going to be greater than zero and therefore, using our little therefore symbol, it always has distinct real roots. Always has distinct real roots.

There we go and we can just write that out. Okay, and that is literally all we need to do to prove that there. Just obviously making that statement that obviously that discriminant there is always going to be bigger than 0, and therefore it's always going to have distinct real roots just using that completed square form that we've done on the previous question.

So there we go, that's how we go about answering these three elements of this particular question here. We had a look at obviously showing the discriminant using these values of k that we had in our quadratic up there. We then obviously looked.

at writing it in completed square form, so expanding and simplifying it and completing the square, and then obviously using that to show that it's always going to be bigger than zero. So that's how we're going to go about answering a question like this, and now here's one for you to have a go at. Okay, so here's your question, so pause the video there, have a go, and we'll go over the answer in a sec. Okay, so for the first thing here, we're going to find the discriminant here.

So we have a equals 1, b is equal to k plus 3, and c is equal to that k at the end there. So if we put this in terms of k for the discriminant, remember when we're doing b squared minus 4ac. There we go.

So that's k plus 3 for b, so we have k plus 3. squared, there we go, and minus 4 times 1 times k. So that's going to be minus 4k. There we go. And that is my discriminant in terms of k. OK, there we are, let's just highlight that.

Right, so for the next part, it says, show that the discriminant can be written in this completed square form where p and q are integers. There we go, so let's expand that out and complete the square. So when we expand it out, and let's just write this down here. so we have k plus 3 squared minus 4k and if we expand that double bracket there we get k squared plus 2 lots of 3k so plus 6k and then 3 times 3 is 9 so plus 9 and then we're also going to minus this 4k at the end.

So again let's simplify that and collect these two terms here and we will get k squared 6 take away 4 is 2 so plus 2k and then plus the 9. Now again we just need to complete the square for that so halving the coefficient of k and we get k plus 1 squared that gives us 1 we want 9 so plus 8 at the end. There we are so that's it written in completed square. form. P is 1 and Q is 8. There we are, so finishing this off, it now says, obviously similar to last time, show that for all values of k, the equation where f of x equals 0 has real roots.

So again, looking at our completed square form here, at the moment we have k plus 1 squared plus 8, and again we're looking at that part in the bracket there. So this part in the bracket, again, is always going to have a value, I'm just going to write always bigger than or equal to 0. equal to 0 and obviously because we've got the plus 8 there even if it was negative 1 subbed into that our value of the discriminant which is b squared minus 4ac is always going to be greater than 0 and because it's always greater than 0 we can say therefore always has real roots There we go, and we can finish that off just by writing that, and there we go, we're done. Okay, so that is obviously all the work on the discriminant that we're going to have a look at.

Obviously we looked at lots of different bits of maths there, so as I said before, I will put the link in the description for work on the quadratic formula. and I'll also put a couple of videos linking to obviously completing the square and some of the harder completing the square types of questions that you can have as well obviously just make sure that you do check those out make sure that you've really clued up on all those topics but there we go that's the end of the discriminant hopefully that was useful and helpful if it was please like please comment please subscribe and I'll see you for the next one

Transcript for:Understanding Discriminants in Quadratic Equations

Transcript for:
Understanding Discriminants in Quadratic Equations