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Hello, I'm Mr Coward, and welcome to today's lesson on Number of Solutions, where we look at the number of solutions for quadratic equations.

For today's lesson, all you need is a pen and paper, or some thing to write on and with.

If you can, please take a moment to remove any distractions, including turning off any notifications.

And if you can, 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 the try this task.

I would like you to solve the following quadratic equations, where possible, and see if you notice anything.

Okay, so pause the video and have a go in three, two, one.

Okay, welcome back.

Now let me talk through my answers here.

So for the first one, what I do is I would subtract five from both sides, and I'm going to write more working out here, and so I'd get two.

Okay.

Then I divide by two, and then square root, and this is where, because I've square rooted, this is where it splits into two solutions.

Take three, take three.

So X equals 6, negative 12.

So there are my two solutions.

This one has two solutions.

Now what about this one, this second one.

I'm going to do a working out here.

So we've got- we take 23 from both sides.

So what would that leave me with? Equals zero.

Interesting.

So then we square root.

Sorry, we divide by 2, we divide by 2 which gives us X plus 1 squared equals 0, then we square root, so we get X plus 1 equals 0, because positive zero or negative zero they're the same.

So we just get 0.

So for this one, we just have one solution, which is x equals negative 1.

Okay.

All right.

Well what about this one? Well, this one, so I subtract five from both sides, I don't know, get negative 28, divide by two.

I'm sorry that I haven't rolled that step in, my pen's playing up so it's just like the easier to have it all right here.

You know what? I'm going to do it.

All right.

I'm dividing by two.

I'll say this is what happens.

Okay.

Now I square root.

Hmm.

Well, if I try, if I try and square root negative 14, you get this on your calculator, if you can see that, you get a math error.

Okay? Your calculator will not let you do a square root of a negative.

I mentioned in the previous lesson that you can actually do this and get an imaginary number, but for GCSE, we do not deal with imaginary numbers.

So you just get no solutions.

Okay? So there are no real solutions.

And that's how we'd say it.

So one of these had two solutions, one of them had one solution and another one had no solutions.

So we don't- So we just ignore that one.

Okay? So, whenever we get a negative and we have to square root that, we will get zero solutions.

We cannot do that at GCSE.

Okay.

So which of these has two solutions, which has zero and- which has two, which has one and which have zero? Well, this one, let's try and work them out.

Well, we square root both sides.

So we get X equals 5 or X is negative 5.

Okay, so you just try and solve it and see if you have to square root a negative.

This one, we try and square root it and we just get, Oh, we just get 0.

There's only one solution.

This one, we try and square root it, but we cannot because we cannot square root a negative at GCSE.

Okay.

We would get imaginary roots which we don't deal with for now.

Okay.

So what does this look like? Well, if you imagine, we've got this graph, okay? This graph is Y equals X squared.

Okay? So we just write that here.

So we've got the graph of Y equals X squared.

Now here, I've got the line Y equals 25, Y equals zero and Y equals negative 25.

You can see that Y equals 25 meets this graph here, which means that X squared equals 25 has solutions.

Okay? Here, you can see the X squared equals zero just touches the graph.

It just touches that line and it just touches it in one place.

So we have one solution.

Whereas over here, they don't touch at all.

And because they don't intersect, because they don't meet, we have zero solutions or no real solutions.

Okay.

So, what about this? Well, here, we've got negative X squared equals 25.

So let's times both sides by negative 1.

And that tells me that X squared equals negative 25.

So this time, even though the 25 is positive, when we get it, so that it's just X squared, we actually get a negative 25.

So this one has no real solutions.

Well, in this one, well, we'll rearrange it.

We'll times by negative one.

Zero times by anything is just zero.

So we get X squared equals zero and we square root that.

So we get two solutions.

So one solution of X equals zero.

This one times both sides by negative one, times by negative 1, we get X squared equals 25.

So then we get our two solutions when we square root X equals five or X equals negative five.

And you can see that from the graph.

So this graph is Y equals negative X squared Okay, negative is going that way.

Okay? Now this is the line Y equals negative 25 and it meets twice.

That would be the point.

That there would be negative five.

And if you drew a line up going from there, that line would be 5.

Equals 0 at one point, and equals 25 never.

So we can see visually from the graph that that side does not intersect with that side at all.

So there are no real solutions.

Okay.

Determine how many solutions each quadratic has.

Well, let's try and solve that.

Take five take five.

Two X squared equals zero, divide by two, divide by two.

Sorry.

So we get X squared equals zero.

Zero divided by two is just zero.

So that means when we square root, we get X equals just zero, just one solution.

This one, a nice one, well, let's try and work it out.

So you get two X squared equals five, divide by two, and we get X squared equals- And I'm going to write it, I'm going to leave it as a fraction.

X squared equals 5 over 2.

So we square root it and we get X equals positive 5 over 2, or X equals negative 5 over 2.

And I forgot to write my X equals there, I'm sorry.

Okay? So there are two solutions.

So this one has two solutions.

So we just, when we're trying to find out how many solutions it has, we just go and try and solve it and see what happens.

So for this last one, I take five.

I take five, I get two X squared equals negative 2.

Alarm bells should be ringing here.

I divide by two, I divide by two and I get X squared equals negative 1.

Well, can I do anything now? No, I can't.

We cannot at GCSE take the square root of a negative.

So we get no real solutions.

So zero solutions, two solutions, one solution.

I'll do it in a circle here.

Okay.

And you can see that from the graph here.

So this is my graph.

Two X squared plus 5 equals- Yeah, that's, my graph is, this Y equals two X squared plus 5.

Okay.

Then this line is X equals 10, sorry.

Y equals 10, not X equals 10.

X equals 10 goes that way.

Y equals 5 and Y equals 3.

And you can see, it touches Y equals 5 at one place.

So it has one solution.

This one, it touches twice.

It crosses twice.

It intersects twice.

So it has two solutions.

This one, it doesn't cross at all.

So it has zero solutions.

Okay? So, sometimes there are two solutions, but it's only sensible to take one.

So for example, this, I think of a positive number.

I want to call it X, I square it.

Then add on three and then multiply by two.

My answer is 206.

What is my original number? So, let's work this out.

Divide by two.

Okay? Subtract 3.

And then I square root, so I get-I'm going to write this over here.

Here we go.

So I get X equals 10 or X equals negative 10 when I square root them.

Now, can they both be my answer? Have a look carefully at the question, or my statement.

This one here, have a look carefully.

I think of a positive number.

So can my answer be negative 10? No, because that is not positive.

So that is not an answer.

It is a solution to this equation but it is not an answer to the question that I'm asking.

So sometimes you've got to pay attention to what the question is actually asking you.

So here, that is my answer.

And if the question said I think of a negative number, then that would be our answer.

But the question says positive number so, it is positive 10.

Okay? So you just need to be careful.

The solutions that we get to the quadratic equation actually answers our question.

Okay, so now it is time for the independent task.

So I would like you to pause the video and have a go.

So pause the video to complete your task and resume once you've finished.

Okay, so here are my answers.

You may need to pause the video to mark your work.

Okay, I've just done that last one.

The question said I think of a one digit number.

19 is not a one digit number so it must be 1.

Okay.

So now, it is time for the explore task.

Now you need to make three quadratic equations with two solutions, make three quadratic equations with one solution and make three quadratic equations with zero solutions.

And try make one of your equations easy, try and make one medium and try and make one hard.

So, pause the video and have a go.

Okay, so I cannot possibly go through all the different solutions for this and all your different creative and wonderful and very imaginative questions but the key idea is, if it has two solutions, you need to at some point take the square root of a positive number.

If it has one solution, you need to at some point take the square root of zero.

And if it has no solutions, you need to at some point take the square root of a negative number.

Now, for this one, we sometimes get others where you don't have to take the square root of zero.

This is just for pure quadratic equations.

So all we've dealt with so far is pure quadratic equations.

And in future lessons, we're going to be looking at where it's not a pure quadratic equation.

Okay? So there's a little tease of fear.

All right.

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.