Lesson video

Welcome to today's lesson.

My name is Ms. Davies and I'm gonna help you as you work your way through these exciting algebra topics.

Thank you for choosing to learn using this video.

The great thing about that is that you are gonna be able to pause things and have a real think if you come across anything you're finding a little bit trickier.

I will help you out in any way I can as we work our way through.

Let's get started then.

Welcome to our lesson on solving complex quadratic equations by completing the square.

You do need to make sure you know the method of completing the square before you get into these really exciting completing the square activities.

So completing the square is the process of rearranging an expression of the form AXE squared plus BX plus C into an equivalent expression of the form A lots of X plus P squared plus Q.

And this is a process that can help us solve quadratic equations.

So we're gonna start looking at how we'd complete the square with complex quadratics.

So let's start by building the expression 2x squared plus 8x plus eight using algebra tiles.

If you've got algebra tiles at hand or a computer programme where you can use algebra tiles, have a play around with this to start.

Right, there's loads of ways to build this expression because we haven't stated what sort of format we want it in.

What we are gonna look at is writing this as two lots of X squared plus 4x plus four.

We have all those terms. We can see that this is two lots of X squared plus 4x plus four.

If we look at X squared plus 4x plus four, we've got X squared plus 4x plus four and we know we want two lots of that.

Just double check, do we have 2x squared plus 8x plus eight? Yes, we do.

X squared plus 4x plus four is a perfect square.

So in this case, we can represent our expression with two identical squares.

Let's just look at what we've done.

We've changed 2x squared plus 8x plus eight into two lots at X squared plus 4x plus four and then written that as the square X plus two all squared.

So you've got two lots of X plus two all squared.

It's really important that you understand what we're doing here before we move on to trickier ones.

So Alex says, "What would happen if the expressions were not perfect squares?" Well, let's try three X squared plus 12x plus six.

We could write that as three lots of X squared plus 4x plus two.

So let's look at X squared plus 4x plus two.

There's X squared, there's our 4x, but we only have two so we don't make a perfect square.

Remember we had three lots of that.

So we'll draw all three.

If we just look at one of those, that can be written as X plus two all squared minus two 'cause we are missing two of those ones tiles in the top right corner.

Let's recap what we did.

3x squared plus 12x plus six is three lots of X squared plus 4x plus two which we can write as three lots of X plus two all squared take away two and it's three lots of that whole thing.

So you'll see that I've got two sets of brackets, one round the X plus two all squared and one round the X plus two all squared minus two 'cause we want three lots of that expression.

I could also write that as three lots of X plus two all squared minus six.

Let's check we're happy where that six comes from.

There are six ones tiles missing from those squares in total, two from each, so six in total.

If we look at the row above, you can see that this is found by multiplying the three by the negative two.

So we no longer need those brackets 'cause we multiplied three by the negative two.

So three lots of X plus two all squared minus six.

Take your time in these early stages to make sure you're really confident with how to manipulate these.

So chance for you to have a go.

Use the diagram to fill in the missing terms in these equivalent expressions.

Okay, so the first one we're missing a six.

If you think about halving two X squared plus 12x plus 12, that'll give you X squared plus 6x plus six.

If you just look at one of our diagrams, you'll see that we've gone X squared, 6 Xs and six ones tiles.

Okay? So then we looked at how that related to our squares, so we're missing a three now.

If you just look at one of those diagrams, you can see we've got X plus three all squared, but we are missing three of the tiles.

So X plus three all squared minus three.

And of course, we've got two lots of those and that means in total, we've got two lots of X plus three all squared, but we are missing six tiles, three from each.

Just pause and make sure you are confident with where those expressions come from before you move on.

Time to have a go at this one.

Same thing as before but there's a couple of extra missing terms this time.

Off you go.

So this time we've divided three by three.

So we get X squared minus 4x plus three.

And if you have a look at one of those diagrams, you'll see that we've got X squared minus 4x plus three.

That can be written as X minus two all squared take away one.

And you can see that in one diagram, we are missing one from our square and that gives us take away three.

In total, there's three ones tiles missing from the entire diagram.

So often it'll be difficult to form squares with algebra tiles as not all the values may be integers.

However, we can use the same method with any expression of this form.

So let's try 2x squared minus 12x plus five.

So if we factor out a two, we can write that as two lots of X squared minus 6x plus five over two.

Well, let's see what X squared minus 6x would look like.

So we'll have an X squared and in order to make a square, we need to half the negative 6x.

So we'll have negative 3x on each side.

There we go.

So we've got two lots of X squared minus 6x but you can see we have a constant of nine in each that we don't want.

So you can write that as X minus three all squared, take away the nine that we don't want, add the five over two that we do want.

And you can see that all of that we have two lots of.

We have a choice now.

We can add these values together and then multiply them by two or we could multiply them by two separately and then add them together, whichever you find easier in each case.

Here, I've decided to multiply them both by two 'cause that will remove the fraction.

So I've got two lots of X minus three squared subtract 18 add five.

Or two lots of X minus three squared minus 13.

Okay, let's try that again.

So I'm gonna do an example on the left-hand side and then you are gonna give it a go.

So I've got 4x squared minus 8x minus six.

I'm gonna start by factorising out the four.

So I have four lots of X squared minus 2x minus 1.

5.

And then I need to half the coefficient of X.

So that'll give me X minus one all squared, but then I have a plus one that I don't want so I need to take away one and take away the 1.

5.

Again I think it'll be easier to multiply those both by four before adding the two of them together.

But you could add them together first and then multiply by four.

So I've got four lots of X minus one all squared minus four minus six or four lots of X minus one all squared minus 10.

And I've now written that in the form A lots of X plus P squared plus Q where A would be four, P would be negative one, and Q would be negative 10.

All right, your turn.

Have a go at this one.

You're gonna want to use fractions instead of decimals 'cause you're gonna have recurring decimals otherwise in this one.

Off you go.

Well done.

Let's have a look.

We've got three lots of X squared plus 2x minus a third, we can half the coefficient of X.

We've got X plus one all squared but then I'm gonna end up with a one that we don't want, three to take away one and also take away a third.

This is gonna be easier to times by three first before adding together.

So we've got three lots of X plus one all squared minus three minus one or three lots of X plus one all squared minus four.

Let's have a look at this one.

So 3x squared plus 3x minus six.

Let's start by factorising out three.

This time we're gonna have a decimal when we half our coefficient of X.

That's absolutely fine.

We've got X plus 0.

5 all squared minus 0.

5 squared minus two.

I am going to square that and add them together before multiplying by three this time.

So that's X plus nought.

5 all squared minus nought.

25 minus two or three lots of X plus nought.

5 all squared minus 2.

25 but of course, I've gotta multiply that by three.

So three lots of X plus nought.

5 squared minus 6.

75.

You might wanna pause the video and just have a look through those lines of working.

Time for you to have a go.

You've got 2x squared plus 6x plus five.

Off you go.

Good effort on these.

There's quite a lot of algebraic manipulation skills that we're bringing together.

So factorising out the two, we should have two lots of X squared plus 3x plus five over two.

Of course, you can write that as 2.

5 if you wish.

Then I need to half the coefficient of X, so you've got X plus three over two all squared and then you need to subtract three over two squared.

Again, it's entirely up to you whether you want to use fractions or decimals.

I think it's easier to square a fraction.

Don't forget, you also need to add on the five over two that you had before.

So you've got X plus three over two all squared minus nine over four plus five over two.

I think it's gonna be easier to add those together and then multiply by two.

But it's your choice.

So you have an X plus three over two all squared plus a quarter and that's all multiplied by two.

So you'll write that as two lots of X plus three over two squared plus a half.

If you wanted to write that as two lots of X plus 1.

5 all squared plus nought.

5, that's absolutely fine as well.

Time to have a practise then.

I'd like you to write each of these in the form A lots of X plus P squared plus Q and you can use the algebra tile or a diagram to help you.

When you're confident, come back and we'll look at the next bit.

So second set of questions.

This is gonna be a lot easier without algebra tiles.

Just using your algebraic manipulation skills.

You've only got three questions to do, so take your time to make sure you've got accurate answers.

Off you go.

Let's have a look at these then.

So I've drawn the algebra tiles for you.

The first one we've got two lots of X plus three all squared minus eight.

B, we've got three lots of X plus one all squared plus three.

And C, two lots of X minus two all squared minus six.

And question two, pause the video and check your lines are working match with mine as well as your final answer.

For C, if you wrote it as a decimal, that's two lots of X minus 2.

5 all squared minus 3.

5.

Check those and then we'll move on to the next bit.

So now what we're gonna look at is solving complex quadratic equations.

So putting those skills together with our solving equations skills.

So completing the square is a useful method for solving quadratic equations.

Let's try 3x squared minus 12x minus 15 equals zero by completing the square.

So let's factor out the coefficient of X squared.

So we've got three lots of X squared minus 4x minus five equals zero and we can now divide both sides of the equation by three.

X squared minus 4x minus five equals zero 'cause zero divided by three is zero.

Now we can complete the square.

So X minus two all squared take away the four take away the five, that's X minus two all squared minus nine equals zero.

X minus two all squared equals nine.

And then solve remembering both solutions.

So we've got X minus two equals three or X minus two equals negative three.

That means X could equal five or X could equal negative one.

Of course, in this case, we could have solved by factorising 'cause they were integer solutions.

However, knowing another method can be really helpful and sometimes factorising isn't going to be easy.

There are different methods to solve quadratic equations.

We're gonna look at 6x squared plus 11x equals 10.

Jacob says, "I'm gonna solve this by factorising." Aisha wants to do it by completing the square.

Let's see if they both work.

So Jacob, he's looking to decompose the X term.

So he wants two values that have a sum of 11 and a product of negative 60.

That's six times negative 10.

That's gonna be 15x and negative 4x.

So we've got 6x squared plus 15x minus 4x minus 10 equals zero.

Factorising that separately, we get 3x lots of 2x plus five minus two lots of 2x plus five equals zero, which is 3x minus two lots of 2x plus five.

One of those brackets has to be equal to zero.

So X is either two over three or negative five over two.

Let's look at Aisha.

She's gonna try and complete the square.

So she starts the same way as Jacob, making sure it equals zero.

Then she's going to divide through by six.

Then she's got a half her 11 over six which is 11 over 12 and then square that and take away 121 over 144.

Not forgetting that she's got a constant of negative 10 over six as well.

Then she's got to add her fractions and rearrange.

So we've got X plus 11 over 12 all squared equals 361 over 144 and that means X plus 11 over 12 equals 19 over 12 'cause we've square rooted or X plus 11 over 12 equals negative 19 over 12.

Don't forget the negative root.

Then we've got X equals two thirds or negative five over two.

Right, I think we can see that Jacob's method was definitely easier.

Where factorising is possible, it often leads to a nicer method than trying something like completing the square.

Particularly as we were dividing through by six and the only one term had a factor of six, the other two didn't.

So do keep your eye out for times where you can factorise.

However, sometimes expressions do not factorise easily.

Completing the square is then a good alternative.

Let's try this one.

2x squared equals three minus 8x.

We want our answers to one decimal place.

So we wanna rearrange to equals zero.

Divide by the coefficient of X squared.

So we divide three by two.

And then complete the square.

So we've got a half our coefficient of X, so we've got X plus two all squared.

We need to take away the two squared so take away four and don't forget we've also got minus 1.

5.

Simplify and rearrange.

So we've got X plus two squared equals 5.

5.

Square root our 5.

5 to get 2.

345 or negative 2.

345.

And then solve.

We wanted our answer to one decimal place so we've got nought.

3 or negative 4.

3.

Time to have a practise.

Follow my steps on the left-hand side and then you are gonna try one exactly the same.

So I'm gonna divide through by the coefficient of X squared.

So divide through by five.

Complete the square.

So I've got X minus three all squared.

I need to subtract the constant of nine that I don't want but add on the two that I do want.

Collect the like terms and rearrange.

I've got X minus three all squared equals seven.

Square root remembering my positive and negative root and then add three.

I want two significant figures.

So I've got 5.

6 or 0.

35.

Have a go at this one on the right-hand side.

Let's see then.

Dividing through by three gives us X squared plus 20x plus 50 equals zero.

Completing the square, X plus 10 all squared minus 100 plus 50 equals zero or X plus 10 all squared minus 50 equals zero.

X plus 10 all squared must be 50.

Square rooting gives us 7.

07 or negative 7.

07 and solving to give us negative 2.

9 or negative 17.

We wanted just two significant figures so negative 17 is fine.

When an equation has a negative X square term, it's helpful to remove a negative factor.

Let's have a look.

So say we had 8x minus two minus 4x squared equals zero.

We could rearrange so the X squared term is first and that might make it easier to see what we're doing.

We want to remove a negative factor so our X squared is positive.

So if we removed a factor of negative four, notice that when we divide positive 8x by negative four, it becomes negative 2x and when we divide negative two by negative four, it becomes positive nought.

5.

So make sure your signs are correct in that step.

Then we can complete the square as normal.

Let's try one together, then you'll have a go.

We've got a fractional X squared term this time, but that's not a problem.

So I'm gonna rearrange so my X squared term is first, just to make this easier for me.

I'm gonna remove a factor of negative a third.

You could do this in two steps.

You could multiply everything through by negative one and then multiply everything through by three.

That would work as well.

So notice that negative five divided by negative a third, that's negative five times negative three, which is 15.

And one divided by negative a third is gonna give you negative three.

So just check you've got your terms correct there.

You can always do a quick expansion to make sure that that works.

X squared plus 15x minus three equals zero 'cause zero divided by negative a third is still zero.

Completing the square, being careful with our decimal values and square rooting to give our two square roots and solving for our two answers.

You wanted three significant figures but 0.

197 or negative 15.

2.

Have a go at this one being particularly careful with your fractional coefficient.

If you need to do it in a couple of steps, that's absolutely fine.

Off you go.

Right, I think this is as mean as it can possibly be.

So well done if you thought about how you were gonna do this, even if you made a few mistakes with your fractions, hopefully, you've got the general process.

So I'll rewrite it so that I've got my X squared term first and I wanna divide through by negative one and I also want to divide through by three over two.

Remember dividing by three over two is the same as multiplying by two over three.

That might help.

So we've got negative three over two lots of X squared minus 6x plus four equals zero.

If you're not sure, just check back that that works.

Then we can divide by negative three over two and complete the square.

Once we square rooted to get our two solutions, we can add three to get 5.

24 or 0.

764.

Don't forget we wanted our answers to three significant figures.

Definitely that first step of dividing by negative three over two was the trickiest in that question.

So well done if you got that.

If you didn't, just a little bit of a practise with your fractional skills will help with those trickiest of questions.

Time for you to have a go yourself.

So I'd like you to solve each of these quadratic equations, giving your answers to two significant figures.

Take your time, make sure your working out is presented clearly and you can follow each of your steps.

That way you'll be able to spot if there is a mistake.

Off you go.

Same again with this second set.

Pay particular attention they're not all equal to zero before you start.

Off you go.

And finally, Aisha is solving 3x squared plus 3x equals 18.

Have a read through her method.

Laura says, "I think she needs to put the equation equal to zero to start." Jacob says, "I think she can solve this by factorising." Firstly, is Laura correct? And explain your answer.

There is at least one mistake in Aisha's working.

So see if you can find that mistake.

And then show that Jacob is correct and get to the solutions using his method.

Off you go.

Let's have a look at our answers then.

So pause the video and check through each of those steps, making sure you are happy with your final answer and where it's come from.

For this second one, make sure for D that you've rearranged to equal zero.

So we need to subtract for X and subtract one from the right-hand side to get zero.

And then be careful with those fractions as you go through.

You get two answers, nought.

15 and negative 2.

2.

For the second one, it might help to rearrange so that your negative half X squared is first, then 2x and minus one.

Or you could have added a half X squared and subtract 2x to get a half X squared minus 2x plus one equals zero.

Doesn't matter which way round you did that.

And then F, negative 2x squared minus 5x plus four.

And then just making sure you've completed the square accurately.

So is Laura correct? Actually, no.

Aisha doesn't need to rearrange the equation equal to zero.

It's okay for it to be equal to a different constant.

It'd be no good if she had X terms on the right-hand side, but having a different constant other than zero is fine.

Because 18 was divisible by three and she started by dividing through by three, this worked quite well.

She did make a mistake later on though.

When she had X plus nought.

5 all squared minus nought.

25 equals six, she needed to add nought.

25.

She did not add nought.

25.

She got in a bit of a muddle there.

So it should say X plus nought.

5 all squared equals 6.

25.

Then she can solve.

Let's look at Jacob then.

Jacob reckons we could do this by factorising.

Let's have a look.

We do need the equation to equal zero for factorising.

We get 3x minus six lots of X plus three.

Equally, you could have divided through by three and that would've made your expression a little bit easier to factorise.

And that gives us 3x equals six or X plus three equals zero.

So X is two or X equals negative three.

Well done for all your hard work today.

You have solved some really complicated quadratic equations.

You should be really proud of yourselves.

We've seen how completing square can be used to solve quadratic equations.

Bearing in mind, sometimes factorising is a more efficient method, but where a quadratic won't factorise easily, this is a nice method for you to use.

When you've got a coefficient of X squared greater than one, dividing each term by the coefficient is a good starting point to make your equation easier to solve.

As with anything, the more you practise, the better you get.

So don't be downhearted if you make some mistakes early on because that means that you'll be able to spot those and improve in the future.

Well done and I look forward to seeing you again.