Lesson video

Hello, I'm Mr. Coward and welcome to today's lesson on solving pure quadratic equations.

For today's lesson, all you'll need is a pen and paper or some things 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 great.

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.

Using substitution, find a value for the expression below when X equals 3 and when X equals negative 3.

Now, I want you to be really careful when you're squaring your negatives here.

So, pause the video and have a go.

Pause in three, two, one.

Okay, so welcome back.

Now, the reason why I said you should be careful with your negatives is because if we have, say we have this, X squared, and if the X was equal to 2, we'd get 4.

Okay, so X squared would be 4.

Well, if X was equal to negative 2, negative 2 times negative 2 gives us positive 4.

So the square of a negative is a positive.

So that's really important.

And when you're, even if you're doing it on a calculator, you need to actually put what your negative in brackets when you're squaring it or your calculator will just think, if you write that, your calculator will just think your squaring the 3 and not the negative.

So if you are doing it on a calculator, you definitely need to put in a bracket.

And if you're not doing it on a calculator, you need to know that a negative times a negative gives us a positive, so a negative squared gives us a positive.

Okay.

So here are my answers.

Now, we've got 3 squared, 9, 9 plus 11, 20.

Negative 3 squared, 9, 9 plus 11, 20.

Oh, that's odd.

They're the same.

11, 11 times 3 squared, so 11 times 9, 99.

11 times negative 3 squared, so 11 times 9, 99, same again.

4 times 9, okay, 3 squared is 9, plus 5, so 4 times 9, 36, 36 plus 5, 41.

4 times 9 plus 5, 41.

Same again.

And on the last one, 5 times 9 plus 1, so 5 times 10, 50, 50 minus 3, 47.

And you can see that it's the same again.

So because we are squaring that negative straightaway and we're not doing anything else with it, we're squaring it straightaway, we get the same answers for 3 and negative 3 because the squaring is the first thing we do.

So that's, that's really interesting.

And that brings us to something that we're going to talk about today.

Quadratic equations can have up to two solutions.

Okay, so what does that mean? Well, here I've got a quadratic equation, and I'm going to solve this.

Now, solving it.

So imagine I'm solving this here, 2X equals 15.

Okay, I divide both sides by 2 to get, my goal is to get just X on it's own or 1X, but we don't actually write the 1, okay? So what we'll do, trying to do is we're trying to find what 1X is.

Now, what's the inverse of squaring? Well, the inverse of squaring is square rooting.

So I square root both sides.

So that gives me X equals the square root of 15.

Not quite.

It gives me the positive square root of 15 or X can be the negative.

Sorry, sorry, the negative square root of 15.

So X can have two values.

It can be the positive square root of 15 or the negative square root of 15.

Because negative 15, negative this times that is also equal to 15.

Just as the positive one, positive square root of 15 times positive square of 15 is equal to 15.

So we actually have two different solutions for X.

And just to introduce you to some notation, some people, I have some very lazy mathematicians will say, don't like to write it out twice.

So some people prefer to write it like this.

Where we have a positive and a negative sign almost in the same sample on top of each other.

For this lesson, I'll just be sticking with this way, but I just wanted to make you aware that sometimes, people write it like that.

Okay, so I'd like you to find the value of X or should I say the values of X? Pause the video and have a go.

Pause in three, two, one.

Okay.

Welcome back.

Now, hopefully, you square rooted both sides.

So hopefully, you got X equals positive root 11 and X equals negative root 11.

Now, just to make you aware, if you did get a surd that could be simplified.

So say you've got this as your answer, and you know how to simplify, well, then, you should.

But if you don't know what simplifying, so it is, and that was just way over your head and you'd never heard that term in your life, then do not worry about it 'cause we will not be including, we will not be focusing on that this lesson.

And if you do know how to simplify it, and you get an answer that could be simplified, then have a go and simplify that.

Okay, X squared equals 25.

So I do my inverse operations.

So X equals the positive square root of 25 or X equals to negative square root of 25.

But I know what the square root of 25 is, square root of 25 is equal to 5, so you've got positive 5 or we've got a square 25, which is 5, so we've got the negative of 5, so we've got negative 5.

So our answer is positive 5 or negative 5 for this one.

Okay, so I'd like you to have a go at this, okay? And find the two solutions for me.

Pause the video and have a go in three, two, one.

Okay.

Welcome back.

Hopefully, you square rooted both sides.

And we know the square root of 36 is 6, so you should have at X equals positive 6 or X equals negative 6.

Awesome.

Okay, so what about this? How is this different from last time? Well, we've got a 2 in front of our X squared.

Oh, we want to get rid of that? How do we that? You can just divide by 2.

And now, we do exactly what we did before, we square root, we square root, so we get X equals, and you know what, I am going to be lazy here, plus or minus the square root of 15.

Okay, your turn.

Pause the video and have a go.

Pause in three, two, one.

Okay, so hopefully, you divided both sides by 3, so get X squared equals 4, then you square root, so you get X equals plus or minus 2 'cause the square root of 4 is 2.

Okay, what about this? Hmm.

Well, I need to get X squared on its own first, so I'm going to do this.

Add 6 to both sides.

That's an awful 6.

It looks like can be it isn't 6.

Okay, and now, we square root both sides.

I'm going to be lazy again, so X squared equals plus or minus 6 'cause the square root of 36 is 6.

Okay, your turn.

Have a go at this one.

Okay, so hopefully, you pause the video and had a go, and hopefully, you did this.

Okay, take 2 from both sides, then square root, so you get X equals plus or minus the square root of 10.

Okay? Awesome.

Let's up the challenge a little bit then.

Okay, what are we going to do first? Are you going to divide by 2? Are we going to square root? Or we're going to add on 4? We're going to add on 4 'cause we're kind of, we're doing the inverse order of operations.

So we subtract 4 last, so that's the thing we get rid of first.

Okay, so because we subtract 4 last, that's the thing we get rid of first.

Then what do we do last now? Well, we square root first, then we multiplied it by 2, so we get rid of the 2 first.

Okay, your turn.

Pause the video and have a go.

Pause in three, two, one.

Okay.

Welcome back.

Hopefully, you subtracted 6 first because that's the thing we did last.

6.

Okay now, what's the thing we do last then? Well, we square root first and then we multiplied by 3, so the thing we did last was the multiply by 3, so we get rid of that first.

And it gives us 2.

So now, to get X on its own, oh, I didn't square root this, I'm very sorry.

So we'd have X equals plus or minus the square root of 17.

So sorry, the positive or negative square root of 17.

Here, we'd have square root both sides, which I'm not going down 'cause I may have another question and we get X equals the square root of 2.

The positive or negative square root of 2.

So there's two answers there, so there's X equals the positive root of 2 or X equals the negative square root of 2, okay? Remember that just because I'm being lazy, I'm writing this plus or negative here, don't forget that there's two solutions, okay? So the two solutions are the positive root of 17 or the negative root of 17.

Here, you have the positive root of 2 or the negative root of 2.

Okay, yeah, and as I suspected, here's my other question, so I'm just going to clear the ink.

So what do we do here? Well, what happens first? So we square root, then we divide by 2, then we subtract 4.

So I'm going to get rid of that subtract 4 first.

Then I multiply by 2.

68, and then I square root it.

So X equals, I'm going to squeeze them a square root there.

X equals the positive square root of 68 or X equals the negative square root of 68, okay? There are two answers.

Okay, so your turn.

Pause the video and have a go at this.

Pause in three, two, one.

Okay, Welcome back.

Now, here are my answers.

I would have subtract 6 from both sides, so we get X squared divided by 3 equals, 6 times by 3, so we get X squared equals 18, and then we square root, and I'm going to write it over here, so we've got X equals the square root of 18 or, the positive square 18 or the negative square root of 18.

And if you know how to simplify, that one does actually simplify, okay? That simplifies into 3 root 2 or negative 3 root 2.

Okay.

What about this one? How is this different? Well, what do we do first? We do what's in the brackets first.

So we do X plus 3 first, and then we square root.

So we actually do the squaring last here, so we get rid of that first.

So here, we've got X plus 3 equals 5, positive 5, or X plus 3 equals negative 5.

So then, we get X equals, X equals 2 or subtract 3 from both sides, X equals negative 8.

Hmm.

This one's a bit different, isn't it? Because now, the last thing we're doing is not the squaring, so we don't have the symmetrical answers anymore.

We don't have the positive and negative version.

So this time we have to write it out X equals this or X equals that, okay? And that we don't have the symmetry because we're, the last thing we're doing, or, yeah, the last thing, what I'm doing is not the square root, it's not the square, so we don't have these symmetrical answers anymore.

Okay, so I want you to pause the video and have a go at this.

Pause in three, two, one.

Okay.

Welcome back.

Hopefully, you square root it first to get this.

Okay, and then add 2, add 2, so we get X equals 9 or X equals negative 5.

And just be careful with your negatives there.

Negative 7 plus 2 is negative 5.

So I just cleared the ink so it's not in the way.

What about this one? How is this one different? Hmm.

Well, this time, we're timesing it by 2 as well.

What's our order? What do we do first? Well, we add the 4, then we square, then we times 2.

So we've got to get rid of the times 2 first 'cause that's the thing we do last.

So we divide by 2, and we get X plus 4 squared, then we square root, so we get our two solutions now.

Every time we square root that's where it splits into two solutions.

So we get X plus 4 equals root 20 or X minus 4 equals the negative root of 20.

So then, I add 4 to, sorry, take 4 from both sides, so here I get root 20 subtract 4 or I get X equals negative root 20.

Yeah, no, sorry.

That should've been a plus 4, so subtract 4.

Sorry, that's a subtraction sign.

It like an add, but it's not.

Let me write it off.

There we are.

Okay, so negative root 20 subtract 4.

So we still get our two answers there.

Okay, so it doesn't matter if it's in that form if we've got a square root, and we just don't combine the surd part and the number part, we leave them two separate.

Okay, like we won't combine an X and a number.

Okay, so we leave our root and the number not in the root, we leave them separate.

Okay, so your turn, so pause the video and have a go.

Pause three, two, one.

Okay, so here, we've got, add 9 first.

16, we square root, so we got 4 or negative 4.

Add 2 to both sides, so we get X equals 6 or X equals negative 2, okay? So we get our two different solutions.

Okay, so now, what I would like you to do is I would like you to have a go at the independent task.

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

Okay, here are my answers or some of my answers.

Now, what was unusual about H? Well, when we square root it, 0.

We only got one solution.

So sometimes, it's possible to only have one solution.

Remember, I said it was up to two solutions.

Sometimes, it can have two, sometimes, it can have one, and sometimes, it can actually have zero solutions, which we'll look at in the future lesson.

Okay, and here's the answer to the final questions.

Okay, so really well done.

If you managed okay on that, that is awesome.

So now, it's just time for the explore task.

So placing any number in the gap, can you find three ways to make the following equation have, A, integer solutions and B, irrational solutions, so like surd solutions.

Okay, so pause the video and have a go and resume once you've finished.

Okay, so welcome back.

Now, here are the first at 10, I think, integer solutions, yes.

And how did I work them out? Well, I just used my square.

So I did 3 times 1 squared, so 3 times 1 plus 7, 10.

3 times 2 squared, so 3 times 4, 12, 12 plus 7, 19, and so on, and so on, and so on until I got to 10.

3 times 10 squared, 3 times a 100, 300, plus 7, 307.

And you could keep going forever, and you could do that for 56, you could that for 94, you could do that for 1,371.

if you really want.

Now, irrational solutions, basically, that is everything else other than this sequence of numbers.

And, so if you'd done 11, if you did 14, if you did 28, but , this number here had to be bigger than 7.

So if it was less than 7, you took it away and you would've gotten a negative number here.

And if you get a negative, and you try and square root that or your calculator give you a math error because you don't get a real number, you actually get what's called an imaginary number, which you'll learn about if you do a level further maths.

Okay? But don't want you to worry about that for now.

We just say that it's got no real solutions.

So if that number was less than 7, you'd have no real solutions.

Oh, and just it's worth mentioning, that you could've also had things that give you fractions, so you could've had 7.

5 or 8 3/4 and various things like that, okay? But that needs to be bigger than 7 in that box.

Okay, so, oh, if that was 7, you would've had 0 as an integer solution, which I actually forgot about.

So I missed one off there, haven't I? You could've also had 7.

Hmm, very interesting.

Okay, so that is all for this lesson.

Thank you very much.

Thank you for all your hard work.

And I look forward to seeing you next time.

Thank you.