Use your graph to find approximate solutions to the equation.

Solving quadratic equations graphically

Key Stage 3

Maths

Quadratic equations (9.10)

Use the graph to find approximate solutions to the simultaneous equations

How many solutions are there to the simultaneous equations below.

﻿Hello and welcome to today's lesson on solving quadratic equations graphically. For today's lesson, all you'll need is a pen and paper or something to write on and with. If you could please take a moment to clear away any distractions, including turning off any notifications that would be brilliant. And if you can, please try and find a quiet space to work where you won't be disturbed, okay? When you're ready, let's begin. Okay, so time for let's try this task. I would like you to solve the following simultaneous equations and see if you notice anything, all right? So be on the lookout. Look at your solutions. Look at the equations. What do you notice? Okay, so pause the video and have a go. Pausing, three, two, one. Okay, welcome back. So hopefully you noticed that this, if you look closely, has a solution of three and eight, that is a solution to the simultaneous equations. This has a solution of three and five. This has a solution of three and two. Okay, what do you notice? The X coordinate is always the same. What else might you notice? The Y coordinate goes down by three. Why might that be? You might want to explore the why coordinate on your own, but I'm just going to focus on the X coordinate. I'm just going to focus on why the X coordinate is the same. So, we're going to look at our equations in more detail. Okay, so when we're solving simultaneous equations, what are we doing? Well, we're finding out when this Y is equal to this Y. So we're finding out when these two are equal and if those two are equal, what else must be equal? These two must also be equal. So four X minus four must be equal to two X plus two. Okay, the Y is equal because they cross, they intersect. When they cross, they have the same Y coordinate. When these two cross they'll have the same Y coordinate. So these two will be equal, which means that these two are equal. When these cross, the Y quadrants are equal and these are equal. Okay, so now we have three equations. And do you notice anything with these equations? Have a look. Okay, what about this? Four X and two X, three X and one X, two X and zero X. Okay, what I'm going to do is I'm going to take one X from both sides. Oh, that's that? Okay, I'm going to take another X from both sides. Oh, that's that. Interesting, and now I'm going to solve this. I'd fall to both sides. Two X equals six divide by two. So that means X is equal to three. So X is equal to three. Where have we seen that before? Well, that was the solution to our three simultaneous equations. So, we can-- We can do the same thing to both equation and get the same answer because when we're solving equations, we can manipulate equations, do the same things to both sides and get the same answer all the same value of X. But the Y doesn't work that way. And unfortunately, we're not going to explore that. Well, it might be something you want to explore on your own. You might want to explore what the solution would be to this. Okay, does that fit the pattern? What about six X and four X? Does that fit the pattern? Okay, so you may want to explore that at another time, but for now we were just establishing the X is invariant, okay? And this is important for today's lesson. Okay, have a look at this. Y equals X squared plus one, Y equals three, Y equals X squared minus two, Y equals zero. What's the same, what's different? Let's have a look at our X solution, negative 1.4. What's our X solution there, negative 1.4. Let's look at our other X solution because there's a pair of solutions, now there's two, 1.4. Well, what's our solution there at 1.4. That's interesting, that's really interesting in fact. Why, why are they the same? Why are they giving us the same X? Well, can you see what we've done here? We've kind of got this, we've got this equation, we've got X squared plus one equals three. And if we take three from both sides, we get that. Can you see that there? X squared minus two, zero. So, the key thing is, is that we can adjust, we can transform simultaneous equations equally. So do the same transformation to both of them. So in this case, we've shifted them down by three and we will get the same value of X, okay? We will get the same value of X and that is really important. And it's really useful as well. Okay, and here's another one. This time it hasn't just been shifted down. Let's have a look at this. That's four point, I think that's-- Yeah, so it's about 4.3 out there. Yeah, about 4.3, something like that. Next one, 0.75. Okay, what if we go over here, 0.75, 4.3. They have the same solutions. Why do they have the same solutions? What have we done? Well, we've had X squared minus three X plus two equals two X minus one. What I've done to this one, is I've shifted it. It's harder to see, but can you see how the lines gone like that to being like that? So it's kind of a weird kind of stretchy rotation that's gone on and you don't need to worry about what transformation it is, but let's look at how it affects the solution. So if I take two X from both sides I also add one to both sides to make that zero, I get X squared minus five X plus three equals zero. And that's what we'd have from over here. So they have the same value of X because it's a transformation to get from one to the other. Now, having it equal to zero is really helpful. It's really useful and it links to something that's called roots and solutions of a quadratic equation, okay? So, been able to shift there and maintain the solutions and get equal to zero is a really, really powerful idea. Okay, so what have we got here? Use the graph to solve this. So this is the graph, Y equals X squared minus four. Okay, let's label it. Okay, now we'll use it to solve that. Well, let's pretend that that's one of our Y's and that's our other Y. So that's like Y equals zero, which is that line there. Okay, that's the access there. So solving it is the same as solving the simultaneous equation of Y equals X squared, plus four X minus four and Y equals zero. It's the same thing as doing simultaneous equations. So it's just this point here, the point where the intersect at negative five, zero and one, zero. So we have our two solutions to the equation, but we're ignoring the Y, because the Y we don't need any more. We don't have any Y's here. So we're just focusing on the X. So instead of writing in the incoordinate form, we're going to cross out, because we're not going to write it like that anymore. We're going to write this as X equals negative five or X equals Y. So we're ignoring the Y. We're not paying attention to the Y because the Y changes and there's no Y in this. We're just trying to solve it. We're just trying to find the value of X. So I don't need to have this Y coordinate anymore. I just need my X's. Okay, now solve this equation. X squared plus four X minus four equals four. Well, that's the same as saying that this line is Y equals four. So I can draw my line, Y equals four here. Don't know why I've drawn a dot, Y equals four. So now, I can find approximate solutions. So that is 1.6, X equals 1.6 or X equals negative 5.6, okay? And if you want to know something cool. Well, I think it's cool anyway. Do you see the line of symmetry is that line. The line X equals negative two. Well, that point is three that direction from negative two. And that point is three that direction from negative two. That point is 3.6 that way and the other point is 3.6 in the other direction. So our symmetry is around negative two this time. So both solutions are an equal distance from negative two, which is really interesting, but just a bit of a tangent. Okay this one, X squared plus four X minus four equals negative six. Well, what are the solutions to that equation? Well, we can draw our line, Y equals negative six, begin with the dotted line to help us. So that would be, if I drew accurately, that would be that point there. When X equals I'll say about 0.3. Sorry, negative 0.3 and this one would be when X equals negative 3.7. So we've got our two solutions X equals negative 0.3 or X equals negative 0.37. And note this, that is 1.7 away from negative two going that way and that is 1.7 away from negative two going that way, which I think is really interesting. And it's all to do with the symmetry of quadratics. Okay, what about this? So we still got our line, Y equals X squared plus four X minus four, but now this time we want to solve a different equation. Plus kind of a nine. Do I have to draw the other graph? And the answer is, no. We can transform our graph or we can transform this equation into that equation. So I'm going to write underneath. I'm going to write this equation here. Equals, how do I get from this to this? Well, the X squared stays the same. What about the four X? Do I need to do anything to that? No, what about this? What do I need to do? I need to subtract six. That gets me to negative four or minus four here. Yeah, subtracting six gets me to the same as subtracting four over here. So if I've done that to that to keep my solution the same, what must I do to this side? I'm must subtract six as well. So now, the solutions to this equation, the X values are going to be the same as the X values over here. So our X values will be again negative X equals negative five and X equals negative one. And if you want, you can use substitution to check that and you will get when you substitute them into that, you will get six. Okay, what about this one? Well, I need-- The thing that I need is I need that on this side, okay? I need that so I'm going to write it beneath, because that's what I want to get to. Well, how do I get from there to there? What operation do I do? Well, it's the same as before I just need to text six. So I text X from the other side, this time I get four, not zero. So it's not that line that I care about, it's this line. So where does it cross that line? Where it crosses there and there. So that's X equals negative 5.6 and X equals 1.6. So they are my solutions, okay? Sorry, that shouldn't have been negative one, that should have been positive one before. I do apologise about that. Yeah, so just be careful with that. Okay so, we've just got to be really-- Yeah so, sorry. We need to get that side equal to the same as our graph and we can transform it and we must do the same to the other part. We must transform that side as well. Then our solutions are maintained. Okay, it's a lot of this, all right? There's a lot going on. So let's try and do another example. Okay, well that's my graph. So my graph is Y equals-- I need to use that graph to solve this. X squared plus four X minus four equals something. Or how do I get from this to this? What do I have to do? I have to take three X and take one. So I do it to that side, I get zero. So, solving this is the same as solving when that equals zero which we've already done and we already know is X equals negative five and X equals negative one, okay? Sorry, positive one, I've done it again and positive one. Okay, hopefully you're shouting at your screams then Mr. Coward, you've made a little mistake. Okay, so here we've got two X squared plus five X minus four equals X squared plus X. Well, what do we need on this side? We need X squared plus four X minus four. Well, how do we get that? Well, we have to subtract X squared, subtract to X and that-- Oh, that's fine. So then we do the same for that side and we get zero again. So our solutions here are the same, positive one. Okay, if so because we can transform it, we've transformed it like that and we've got the same solutions. All right, last one. Here we've got three X squared minus two X minus five. So when does that equals zero? Well, that's easy, that's-- Oh, well that's not easy, but it's such more straightforward. It's just when that line, that graph intersects that line, Y equals zero. That's at the point, negative one. And what do you think about that 1.7 or 1.6, X equals negative one and X equals I'm going to say-- Yeah, I'm going to say 1. 1.65 split the difference. Okay, so then my two solutions. Now, how can I use this graph to solve this? Well, let's write the equation of my graph underneath. Well, what do I have to do? Well, I have to take away two X squared and I need to take away an X to get from negative X to negative two X. So, on this side take away two X squared, zero X squared and take away X, which just gives me zero. So the solutions to this is the same as the solutions to this. Okay, awesome. All right, what about this one? Well, what do I need to do? I need to get to three X squared minus two X minus five. How do I do that? Well, I have to add on an X squared? So I add on an X squared over here and oh, I also need to add on a-- So, I need to take away two. So I need to take away two. So that will mean I just get negative two on that side. So that is when does that meet this line? So my solutions will be X equals 1.4 and X equals negative 0.7. Negative ,0.7. Okay, I need you to check they should both be the same distance away from that point because of assymetry. Okay, final. Oh yeah, no. Now, it's your turn. Okay, so I would like you to have a go. There's three questions, so part of it is to complete your task and resume once you've finished. Okay, welcome back. Here are my answers. Now, obviously you were doing approximate solutions when you're drawing some of the lines in, so your solutions might not be exactly the same as mine, okay? And that's okay. But you should have got the two and the three the same as mine because you can see that that's how it worked, that's where they cross. Okay, so again, some of these are approximate solutions, so yours might not be exactly the same. And finally, okay just on that last one, you would have had to draw the line, Y equals X in, which would have got there like that and then it would have met there and somewhere over here. Okay, so that is all. Well, that's all for the independent task and now it's time for the explore task. So this is what your graph shows, Y equals X squared minus six X plus three. And you need to create three equations that you can solve using this graph. Make one easy, one medium and one difficult, okay? So create three equations that you can solve using this graph. So pause the video to complete your task and resume once you've finished. Okay, so now I can't possibly go through them all, but I'll make a nice, easy one. There we are. That is an easy one. That is probably the easiest one, where it equals zero. or another one that's or you might think it's hard, but I think it's not too bad. It's one of the easier ones, equals two. Okay, that's slightly easier. Or you could have gone one equals negative six where it's got one solution. So something where you don't have to rearrange is probably in the easy category I think. Then in the medium category, you could maybe have one where you had to rearrange something. Lets not omit that. Lets omit that, I don't know, seven and that could be 10, okay? And then you have to rearrange to get X squared minus six X plus seven equals some number, so some linear graph. And I would say in the hard category and you might disagree. You might think this is in the medium category, but something where you rearrange it and you get a line on one side so you might get something like X squared minus six X, after you've rearranged it equals like two X plus one and then you have to plot the line two X plus one. So I would say that's in the hard category. Okay, so that is all for this lesson. Thank you very much for all your hard work. I look forward to seeing you next time, thank you.

Solving quadratic equations graphically

Solving quadratic equations graphically

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UnitMaths / Quadratic equations (9.10)

Maths