Lesson video

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

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

If you can, please take a moment to clear away 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.

Using substitution, find the values of X that satisfy the equation.

And I've said values because there's two for each one.

Okay.

Now the first one, I reckon you'll be able to find the second one.

It's a bit tricky, but give it a go.

Okay.

So pause the video.

And have a go in three, two, one.

Okay.

Welcome back.

Now, here are my two different solutions for this one, this first one we could have got X equals two or negative nine.

And for this one we could have got X equals five or X equals negative seven.

And I think that some of yours might have got these, but these ones less so, and the reason why because using substitution, it's not an efficient way to solve this.

You can solve it.

Sometimes one solution is quite easy to find, but sometimes the other can be really difficult.

I expect that most people probably didn't go and try negatives.

And if you did well done, but the my point here was not to find them all.

But just to show you that it's, it's not the best way to use substitution in some situations.

So I'm going to in this lesson, show you how to solve this one.

And the next lesson, I'm going to show you how to solve this one.

Okay.

So how can we solve this? Well, we can factorised it to make it easier.

So let's, factorised this expression.

So what are the factors of 18? We've got one and 18, two and nine and three and six to four and five.

Not four and five.

And then we get back to six so we can stop.

So here's, here are my factors of 18, but it's negative 18.

So that means we've kind of got two different options.

So we can have the first, these numbers being negative, or we could have these second numbers being negative.

So we could have one and negative 18, two and negative nine and three and negative six.

Okay so, which ones add up to give us seven X.

It be these two.

So, cause nine subtract two gives us seven.

So we'd have in our brackets, we'd have X plus nine, X minus two equals zero.

So now we could solve this.

Now we could say that X plus nine equals zero or X minus two equals zero, and then solve that.

which were our answers from before.

Okay So, as you can see, we've got the answers from before, but I just think this way is easier and it's just more efficient, more effective.

And once you get used to it, you'll be, you'll be really good at it.

Okay so, first step we factorised and then we solve each bracket like we did last lesson.

And so what is the difference between pure quadratic equations and adfected quadratic equations.

Have a look, here are some pure equations.

Here are some adfected quadratic equations.

What can you see any differences? Well, in general.

Adfected equation has a squared and and just a normal one.

Okay so, the power one of that same variable? So power two and a power one of the variable where pure quadratic equations just have a squared and then a constant.

Okay these can have constants too.

Sometimes they do.

Sometimes they don't.

This one wouldn't have a constant when it's rearranged.

So it's not too important the differences is because there's sometimes there's sometimes adfected quadratic equations that it's best to solve like pure quadratic equations.

And there's sometimes pure quadratic equations that you might solve, like adfected quadratic equations.

So don't get too caught up in the differences, but I just kind of wanted to introduce you to them.

Okay.

So we are going to use this principle to solve a quadratic equation.

So what was the first thing we do? Do you remember? We factorised.

So let's factorised this, my factor is a 15, sorry, not two, three and five.

Okay, which two of them add to give us eight.

That would be the three and the five.

So we'd get X plus three, X plus five is equal to zero.

So what of those brackets is equal to zero? So then we get this.

Okay.

So you also, pause the video and have a go pause in three, two, one.

Okay.

Welcome back.

Now so here we should have had a factors of 12 one and 12, two and six, three and four, which two add up to get seven? three and four So we've got.

So one of our brackets equals zero.

So when we solve each equation separately and we get our two answers.

Okay.

So, same principle, just slightly tricky one to factorise, so our factors of 15, but this time, because it's a negative 15 that tells me that they have a different sign, so we can have one and a negative one and 15 one and negative 15, three and negative five, five and negative three.

Okay, which two of them would add up to get negative two? This one, three plus negative five is negative two.

So in our brackets we have this.

Okay, it doesn't matter which way you write the brackets.

Okay.

So if I draw X minus five X plus three, that would have been the same.

We would get the same solutions.

Okay.

So there are my two answers.

So I'd like you to have a go.

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

Okay.

Welcome back.

Now.

Hopefully you factorised it like this X plus 12, X minus one equals zero.

So therefore X equals negative 12 or X equals positive one.

Okay.

And I just sti skipped a step there and skipped a couple of steps to be honest.

Okay.

What about this one? Well, we do the same thing.

We factorised.

So this time it's positive 15.

So that tells me that the two things have the same sign.

Okay.

So there either both positive or the both negative, but they can't both be positive because they add up to get negative 16.

That means they both must be negative.

So which two would it be? X minus one, X minus 15.

So that equals zero.

So X equals one or X equals 15.

Okay.

So there we have our two solutions.

Okay your turn.

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

Okay.

So hopefully you factorised this correctly and you got X minus six, X minus two.

Okay.

They had the same sign and they are both negative.

So we get X minus six equals zero or X minus two equals zero.

So we have that as our our solution set.

And I didn't write that step in there, which was kind of not this step in there, which was kind of not of me.

I should have done that.

So I apologise.

Okay.

Now this one, is this a pure quadratic equation or is it an adfected quadratic equation? Well, it's a pure quadratic equation, but we can solve it like an adfected quadratic equation.

So what we can do is we can say that this cause that factorises to that difference of two squares.

They times together to get negative 16 and they add to get zero.

So negative four plus four adds to get zero.

So would just factorised in that now.

Okay we factorised using the difference of two squares.

The difference of two squares basically says that we don't get next step.

So we can have that.

Oops, sorry.

So X, let me write in the proper step that I should have X minus four equals zero or X plus four equals zero.

So X equals four X or equals negative four.

Now.

Have it your turn.

And I'll explain something in a second.

So pause the video and I've got pause in three, two, one.

Okay.

Welcome back.

Hopefully you use the difference of two squares to factorised that, or just look for two numbers that multiply, add together to get zero X and multiply to get nine.

Okay.

So we have that as our solutions.

So hopefully you've got that.

Now we could have also solved this a different way.

So just I'll do it with this one.

We could have also added on nine to both sides and then square root it.

And we would've got X equals positive three and negative three.

So we would've got the same set of solutions as we got here, but we just did it in a different, in a different way.

And which way is better.

I dunno.

What do you think? You may have a preference.

You may not, but I just want to show you that there's two different ways that we can do this question.

Okay, I am going to clear my screen now.

So what has changed here well, is an X with the 16.

So let's solve this.

We're going to factorised this now because there's no constant term here.

There's a factor of X in common.

So this actually factorised is into a single bracket.

So we get that.

Now we can solve this, that equals zero or X equals 16.

Okay.

So that equals zero or X minus 16.

In fact, let me write that step in X minus 16 equals zero.

So X equals add 16 to both sides.

X equals 16.

And All right.

That step in as well, that's 16.

Okay.

So we can factorised that, but because there's no constant term, we just actually factorised that into a single bracket.

So we get this and then we can solve that by the first thing equals zero or the second things equals zero.

Okay.

So you turn, I'm pause the video and have a go.

Okay.

So hopefully you pause video and have a go.

And hopefully you factorised this correctly into as single bracket.

Okay.

So that factorised into that.

So we're still doing our first same first step.

We're still having to factorised it.

And now because it's factorised either that equals zero.

Or that equals zero.

So one of those two things most equals zero.

And now, we have that.

Okay.

So X equals zero X equals negative nine.

Okay.

So now it's time for your independent task.

I would like you to have a go and so pause the video to complete your task and resume once you're finished.

Okay.

So here are my answers, you may need to pause the video to mark your work.

And now I threw in CND here because they've got they're the same.

They're just written differently that if you add four to both sides, on obviously it's different variables, but that's, doesn't really matter too much.

If you add on four to both sides, they are the same thing as each other, but you might have solved this using difference of two squares.

You might have solved this.

Like you would solve a pure quadratic equation in this case.

I think it's maybe easier to solve as difference to have two squares.

And this case it's definitely easier to solve as a pure quadratic equation, but it's, it's up for debate, you know, you might have done it in different way.

Okay.

So now it's time for the explore task.

And I actually, I couldn't decide between explore task.

So I've got two for you here.

Okay.

So, solve both quadratic equations, what do you notice? Can you find another set of equations like this? or this one, note solve both quadratic equations.

Notice how they are both solvable by factorization, no matter the sign at the constant.

So that sign there, the sign next to, or with the constant.

So that one's positive and that one's kind of negative.

Can you find any other set of equations like this? Hmm.

So they are two explore task.

You can have a go at one of them.

The first one is easier.

The second one is harder.

I struggled with the second one quite a lot, but yeah, just have a go have a go at least one of them, or have a got both of which is probably the best situation and resume once you've finished.

Okay.

So here are my first set of solutions for the first explore task.

So what, what was special here? What did you notice? Well, both of them had just one solution.

And you could have found others.

So how could you have found these well it's anything that squared, so you could have done in this one and this case it's X plus seven squared.

This case is X plus eight squared.

This case it's X minus 10 squared.

Now you might notice something as you start to generate these questions.

Look at this, look at the B term, the, the constant with the X, the coefficient of X.

Do you notice anything there 10 squared, eight squared, seven squared.

That term, when it when was square it, it is just double.

So that's double seven that's double eight and that's doubled double negative 10.

So that's quite an interesting part that you could have noticed and you could have had a play around with, On this one and this one was really difficult.

I actually only managed to find two other examples like this.

I suspect there's more, but I haven't managed to find them yet.

So maybe that's something you did.

Maybe you found a few of them.

Okay.

So that is all for this lesson.

Thank you very much for all your hard work and I hope you enjoyed it.

And I look forward to seeing you next time.

Thank you.