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

Let's get started, then.

Welcome to today's lesson on solving quadratic equations by completing the square.

By the end of this lesson, you'll be confident in solving quadratic equations algebraically by using the method of completing the square.

So that's our key phrase for today.

Completing the square is the process of rearranging an expression in the form, axe squared plus bx plus c, into an equivalent expression of the form, a lots of x plus p squared plus q.

This is gonna make more sense throughout the lesson.

This is a method that can be used to solve quadratic equations.

So let's get started.

We're gonna start by exploring this concept of completing the square.

So what expression is represented in the diagram below? Have a look.

How could you write this as an expression? So Jacob's gone for x squared plus 8x plus 16.

Do you agree? Jun has gone for x plus 4 times x plus 4.

And Laura says it's a perfect square, so I wrote x plus 4 squared.

All of these are equivalent.

I wonder if any of those were the ones you came up with? So which of these show an expression that could be written in the form, x plus a, squared? X plus something all squared? Have a look at the diagrams. Have a think about what expressions they represent, which can be written as x plus x squared.

Off you go.

Well done if you said the first one and the third one.

We're gonna look at these all individually.

The first one is x plus 3, all squared.

One dimension is x plus 3.

The other one's x plus 3.

Now this one was a bit tricky.

Although the algebra tiles form a square, it's not a perfect square, as the side length are different.

One of them is x plus 3 and the other is x minus 3.

So although we're representing them as the same length, because one of them has got a negative term, they're not the same expression.

This one though, x minus 3 all squared can be written in that form.

And the last one, x plus 1, x plus 4, that is not a perfect square.

You can see with our algebra tiles, we haven't got a square, and the side lengths are different.

So we can use algebra tiles to see which expressions can be written as perfect squares.

So here's three examples.

If you have your algebra tiles to hand or a computer programme with algebra tiles, you might want to see how you can build these.

So the first one.

We've got our x squared tile.

We've got 2 x's.

So let's put one on either side of our x squared, and then plus 3.

Well there's plus 1, it doesn't work with plus 3.

X squared plus 2x plus 1 would form a perfect square.

It would be x plus 1 all squared.

Okay, let's look at the next one.

So you've got x squared, we've got 4x, so let's put two there and two there.

And then we need to fill in the rest of our square, and that takes four tiles.

So x squared plus 4x plus 4 can be written as x plus 2 all squared.

Let's try x squared plus 6x plus 6.

So there's our x squared.

Let's do 3x and 3x.

And then to fill in our square, oh, six doesn't quite do it.

We would need nine to fill in our square.

So x squared plus 6x plus 9 would be x plus 3 all squared.

But x squared plus 6x plus 6 does not form a perfect square.

Let's see if we're happy so far.

So I'd like you to write an expression for each diagram in the form x plus a squared.

So we should have x minus 1 squared.

X minus 3 squared.

And this one, x plus 5 all squared.

Okay, so which of these can be written in the form x plus a all squared? See if you can see any structures, any elements that make it easy to tell.

Off you go! Okay, well x squared minus 36 is the difference of two squares.

So not a perfect square.

'Cause one of the binomials has a positive six term, and the other binomial has a negative six term.

Well, and if you spotted this second one works, because it's x plus 6 all squared.

Next one is also going to be a perfect square, because it's x minus 6 all squared.

The bottom one though does not factorise with integer values.

We can use this general form to write expressions which are not perfect squares.

Let's have a look at how this will work.

This is the diagram of x plus 4 all squared.

So this one would represent x plus 4 all squared minus 2.

You can see from the diagram that we are missing two ones tiles.

So we could write it as x plus 4 all squared, take away the two missing ones tiles.

How could we write the expression, x squared plus 8x plus 10? What do you think? We wanted x squared plus 8x plus 10.

So we could have x plus 4 all squared, like we do in our diagram, but then we need to take away six of those ones tiles to get 10 instead of 16.

So what we've played around so far is this idea of completing the square.

So completing the square is a method where we take an expression in the form, axe squared plus bx plus c, so a quadratic trinomial.

And we rearrange it into an expression of the form, a lots of x plus P squared.

So this idea of having a bracket squared plus Q, where Q is a constant.

For this lesson we're gonna focus on quadratics with an x squared coefficient of one.

So something like x squared minus 6x plus 8.

If you've got algebra tiles, see if you can use those algebra tiles to build this, making a shape which is as close to a square as possible, thinking about what you are doing as you go through.

Pause the video and then we'll look through it together.

So there's our x squared tile.

So our x tiles, we will need negative three.

So three negative x tiles on each side.

There we go, so we've got negative three and negative three.

The reason why that worked is it was half of negative six.

We need them to be the same in order to make a square.

So if we halve our negative six, we'll have negative three on both sides.

Now if we completed the square, what would the constant term be? If we filled this in so it looked like a square, what would our constant term be? What do you think? Right, it would be nine, positive nine, because negative three times negative three is positive nine.

But we didn't want positive nine.

We wanted positive eight.

So we could write our expression as x minus 3 all squared, take away one.

Laura says, "I think x squared minus 2x plus 8 can be written as x minus 1 squared plus 7." Use algebra tiles to see if Laura is correct.

Let's have a look then.

So we've got x squared minus 2x, so we put one on each side of the square.

We've got one there to make our square, but we wanted eight.

So we need to add an extra seven, so Laura is correct.

X minus 1 all squared is x squared minus 2x plus 1.

We wanted x squared minus 2x plus 8, so we need to add 7.

Fantastic.

Let's see if we're happy so far.

So this diagram represents x plus 2 squared.

Write an expression for each of these in the form x plus p squared plus q.

Use that diagram in the top right hand corner to help you.

Off you go, and then we'll see if we've got this.

So the top one, we are missing one of the ones tiles, so we can write it as x plus 2 all squared minus 1.

The second one, x plus 2 all squared minus 4.

And then x plus 2 all squared plus 2, we've got two extra tiles.

Now this one's a little bit different, 'cause this is x minus 2 all squared, and then take away three missing ones tiles.

Time for you to have a practise.

I'd like you to write each of these in the form, x plus p squared plus q.

You could use algebra tiles or a diagram to help you if you wished.

Then I want you to think about what you notice about your value for p.

Come back when you're ready to look at the next bit.

Well done, so this time we've got some statements.

I'd like you to tell me if they're true or false.

Again, you could use algebra tiles to help you, or you can think about your algebraic manipulation.

Once you're happy with which ones are true and which ones are false, come back and we'll explore this together.

Fantastic.

So I've drawn the algebra tiles for you.

X plus 1 all squared plus 1.

For B, we've got x plus 3 all squared, minus 2, 'cause we should have nine, we only want seven.

C, we've got x minus 2 all squared minus 1.

And hopefully you started to spot that our value for p is always half the coefficient of x in the original quadratic.

So for A, we had 2 x's, we halved that, and that was our value for p.

For B, we had six x's, halving that gave us our constant in the bracket, so our value for p.

and the same as C, we had negative 4x, halving to get negative two gave us our value for p in our bracket.

So let's have a look at these.

So x squared plus 4x plus 2 is x plus 2 squared minus 2.

That is true.

B, this one is false.

We've just talked about how you need to halve the coefficient of x to know what's gonna go in your bracket.

'Cause remember we're trying to make a square.

We need to half the coefficient of X so we can put them on both sides of our square.

C, that's not gonna be equivalent, because if you think about what negative three all squared is, that's gonna give us positive nine, not negative nine.

So that is not a perfect square, that one.

D is correct.

We've halved our negative 8x, so we know that there's gonna be negative four on both sides of our square.

And that would give us 16, but we only want six, so we'd have to take away 10 ones tiles.

E, this is incorrect.

If I had x plus 5 all squared, that'd give me x squared plus 10x plus 25.

For F, if we think about x plus 3 all squared, our constant is gonna be positive nine.

We've actually not got a constant at all on the left hand side, so we need to take away nine on the right hand side to make those equivalent.

Starting to get tricky now, well done if you're spotting these.

G is true.

X minus 1 all squared would give you x squared minus 2x plus 1.

We want x squared minus 2x minus 1, so we need to subtract two.

And finally x minus 12 all squared would give us x squared minus 24x plus 144.

We need to take away 100 if we want it to be equal to 44.

Well done, you've spent a good amount of time there playing around with this new format.

So now we're gonna use it to help us solve equations.

So writing an equation in this form can make it easier to solve.

How can we solve the equation, x plus 3 all squared equals 16? Well let's have a look.

Because I've got a bracket squared equals 16, I can start by square rooting both sides.

If I square root the left hand side, I get x plus 3.

If I square root the right hand side, I've got the square root of 16.

Now notice that every positive number has two square roots, a positive and a negative, which is why in front of my root symbol, I've written a plus and a minus.

That's to tell me that I could have the positive root of 16, or the negative root of 16.

So it gives me two options.

One option is that x plus 3 is 4.

The other option is that x plus 3 is negative 4.

I need to use those to find my two solutions.

Let's look at a graph to help us check.

So here's the graph of x plus 3 all squared.

And you'll see that it intersects the line Y equals 16 at negative 7, 16, and 1, 16.

So those are our two solutions.

So by writing quadratics in this form, it allows us to solve equations using inverse operations.

So let's try solving x squared minus 4x minus 5 equals 0.

So to form a square, we need to half the x term.

So we'd have x minus 2 all squared.

But remember that would give us a constant of 4, which we don't want.

So we can subtract 4.

So x squared minus 4x would be x minus 2 all squared, subtract 4.

But we do want a constant, we want the constant negative 5.

So we also need to add negative 5.

So we've got x minus 2 all squared.

Subtract the constant of 4 that we didn't want in our square and then add negative 5.

And remember our equation was equal to zero, so we still need to have equals zero.

Now we can collect like terms, so we've got x minus 2 all squared minus 9 equals zero.

X minus 2, all squared equals 9.

And this is the key bit, there are two roots to the value 9.

We've got positive three and negative three.

So we need to solve these separately.

So x minus 2 is 3, or x minus 2 is negative 3.

That means x could be 5 or negative 1.

Again, we could check on our graph.

So we've got x squared minus 4x minus 5.

And we can see that it crosses the x axis, 'cause it's equal to 0, at negative 1 and 5.

Laura says, "Wouldn't it have been easier to factorise and set the binomials equal to zero?" What do you think? Course, Laura's correct.

For this example, it would've been easier to solve by factorising.

It's equal to zero, it factorises nicely, as x minus 5x plus 1, and that would've been our quickest way.

However, completing the square is useful when a quadratic does not factorise with integer values.

So let's look at this one.

X squared plus 16x plus 44 equals 0.

You could spend ages trying to look for values that would work, and you would notice that this will not factorise using integer values.

So let's use this method of completing the square.

If you'd like to give this a go on your own, pause the video and try it now.

Otherwise, we'll go through some steps together.

Okay.

So x squared plus 16x.

If we're making a square, we need to half that 16.

So that gives us x plus 8 all squared.

If we think about that, that's gonna give us 64, that we don't want.

So we can take away 64.

but we did want 44, so we can add back on 44.

Of course you could do that in one step, and realise that we had 64, we want 44, so we need to take off 20, that's fine as well.

So you can either take off the whole constant and then add on what you want, or you can work out what the difference between the two are.

I find that first method a little bit easier to not make mistakes.

So there we go, that's our collecting like terms. X plus 8 all squared minus 20 equals 0.

And then we can solve using inverse operations, so add 20 to both sides.

X plus 8 all squared equals 20.

But Sophia's not sure about what to do now.

"I don't know the square root of 20." And I think that's pretty fair, Sophia, because off the top of my head, I don't know the exact square root of 20 either.

So we can use our calculators.

So use your calculator for the square root of 20.

And then we want it in decimal format, so if we press the format button, scroll down to decimal, and select that, we get our answer as a decimal.

Different calculators will have different ways of changing your answer to a decimal.

Laura's thinking.

"The calculator has only given me one root.

I thought there was always two." Yeah, she's right, isn't she? If you use the square root button, your calculator will give you the positive root.

But of course there's also a negative root.

So when we write our values out, we've got x plus 8 is approximately 4.

47, 'cause we've rounded our values, or x plus 8 is approximately equal to negative 4.

47.

Unless directed otherwise, choose a degree of accuracy that sounds sensible.

I've gone for three significant figures.

And then we need to do the same operation to both sides to maintain equality.

We subtract eight, we get negative 3.

53 and negative 12.

47.

Try this one and then I'll give you a go.

So we've got x plus 5 all squared.

So that's gonna give me x squared minus 10x plus 25.

So I need to take off the 25, but add on the 10 that I wanted.

Collect like terms, and add 15 to both sides.

Now I can square root 15, remembering I want the positive and the negative root, and round to a sensible degree of accuracy.

Our answer's to three significant figures, so we need them to at least three significant figures.

You might want to use four significant figures.

The important thing here is whatever you write down, to keep your full answer in your calculator before you do the next step.

So don't round your answer off and type it back into the calculator.

Use the exact answer in your calculator, then we can add five to it, which gives us 8.

87.

And then the same to our negative value to give us 1.

13.

Time for you to have a go.

Give this one a go and then we'll look at the answers.

So halving our coefficient of x gives us x plus 3 all squared.

But that would give us x squared plus 6x plus 9.

We need to take off the nine.

And we also need the negative five that we wanted.

Collecting like terms, and rearranging, we've got x plus 3 all squared equals 14.

So that means x plus 3 is both the positive and the negative root of 14.

So if we subtract three from both sides, you get 0.

742.

And it was important that we'd kept our full answer in our calculator for that one, 'cause you can see that we need three decimal places as three significant figures this time.

For our other answer you get negative 6.

74.

So Sophia is trying to solve x squared plus 7x plus 2 equals 0 by completing the square.

"I'm not sure I can do this one as seven is an odd number." What do you think? Right, we can solve it in exactly the same way.

We're just gonna have some fractions or decimals in our working, but that's okay, we can deal with fractions and decimals.

So we need to half our coefficient of X, so we're gonna have 3.

5.

You can leave it at seven over two, or use 3.

5, whichever you find easier.

Generally, fractions are easier to manipulate than decimals.

So what would the constant term in our square be? Well, it would be 3.

5 squared or seven over two squared.

So this is what we need to subtract.

And I'm gonna write that as 49 over 4.

But then of course remember we wanted a constant of two to add our two on.

And this may be complicated, but remember we're allowed to use calculators here, 'cause we're gonna have to square root using a calculator anyway.

So a bit of rearranging, and we get x plus 7 over 2 equals the square root of 41 over 4.

We didn't have an instruction of what to round our values to, so we've gone for two decimal places.

So I'd like you to apply the exact same process to these ones.

I'm gonna show you how to do it on the left hand side, and you are gonna have a go on the right.

So we've got 1x, so to half that we'll have 0.

5.

So x plus 0.

5 all squared.

But of course we are gonna have 0.

5 times 0.

5 as a constant that we don't want.

So I'm gonna subtract 0.

5 squared.

Not forgetting that I do want a constant of negative 8, subtract 8.

So a bit of rearranging, and I have x plus 0.

5 all squared equals 8.

25.

Square root gives me positive 2.

87 and negative 2.

87.

Make sure you are using the exact values in your calculator so you don't have any rounding errors, which gives us 2.

37 and negative 3.

37, to three significant figures.

There we go.

Time for you to have a go.

Absolutely use your calculator to help you, and follow the lines of my working on the left hand side.

Off you go! So we've got x minus 1.

5 all squared.

So we need to take away 1.

5 squared, and we need to add on one.

Square 1.

5 and collect like terms. We've got x minus 1.

5 all squared equals 1.

25.

Square root.

So our solutions are 2.

62, and 0.

382.

Well then if you got those, if you're slightly out, particularly with that second solution, it's possibly you rounded your answer too much in your calculator for adding 1.

5.

Time to have a practise then.

So for 1a, we are solving x squared plus 10x plus 24 equals 0.

And I'd like you to fill in the gaps for this method to solve by completing the square.

One solution is given to you.

You do need to find the other solution.

For B, this time we're solving x squared minus 8x plus 8 equals zero.

Both solutions are there for you.

You need to fill in the gaps in the method.

Off you go.

For question two, time for you to have a go on your own.

Can you solve these, giving your answers to two decimal places? Off you go.

A few more to do, you're gonna have to be careful with your fractions and decimals this time.

By all means leave your answers as fractions.

Or if you prefer working with decimals, that's fine as well.

Your final answer though, needs to be to two decimal places.

Use your calculator to help you.

And finally, one of the questions in Laura's textbook is this.

Solve x squared plus 6x plus 10 equals zero.

Laura's done some steps of working.

Then she says, "I keep getting an error message on my calculator when I try to do the next step." What step is it that Laura's trying to do, and can you explain why she's getting an error message? What does this tell you about this equation? So a little bit of thinking and reasoning with this one.

When you are happy with your final answer, come back, and we'll look at the answers together.

First one, so we need to fill in the missing gaps in our working.

To start with, we need to subtract 25, to then add on our 24.

Or, that's the same as subtracting one.

Then we've got x plus 5 all squared equals 1.

So x plus 5 equals 1, or negative 1, which gives you x equals negative 4, or the other solution, x equals negative 6.

You can see that that one would have factorised as well.

So B, we need to fill in our steps.

So in our bracket we should have x minus 4 all squared.

We need subtract 16 because it's 4 squared, and add 8.

Or that's the same as subtracting 8.

That gives us x minus 4 all squared equals 8.

And we've got the approximate roots of 8, which is 2.

83 and negative 2.

83.

And that should give us those solutions.

Make sure you're happy with those methods, and then we'll have a look at the ones you did on your own.

Fantastic.

I'd like you to take the time to have a look through these methods.

I'm gonna draw your attention to particular bits.

So for question two you should have got x equals negative 2 and negative 4.

This one you could have solved by factorising instead of completing the square, if you wished.

B and C, however, you would need to use the completing the square method.

You should get 1.

24 and negative 3.

24 for B, and 12.

39 or 1.

61 for C.

For D, E and F, it was up to you whether you use fractions or decimals.

Again, pause the video and check your working out.

Our final answer we wanted to two decimal places, so negative 0.

21, negative 4.

79.

And then 9.

11 and negative 0.

11.

2.

79 and negative 1.

79.

Again, if you weren't confident with those, just take some time to check back through your working, see where it differs from mine.

So Laura's dodgy textbook question then.

So the reason she got an error message was 'cause she was trying to square root negative one.

You cannot square root a negative number and get a real value.

So Laura's calculator would not be able to do that.

That means Laura will not be able to solve this equation.

There's nothing wrong with her working out.

If you check her working out, it works perfectly.

What this means is x squared plus 6x plus 10 equals 0 does not have any real solutions.

If you drew the graph, it would not cross the X axis.

So the textbook obviously made a mistake when writing the question, 'cause it's not solvable.

Well done.

That was a new thing that we looked at today, which is a really exciting method that we can use to solve equations in the future.

We have looked at how completing the square can be used to solve quadratic equations, and the fact that this is most useful when they cannot be easily factorised.

It allows us to solve equations using inverse operations.

And the key thing we've gotta remember is that we'll get two answers when we square root a positive value, a positive root, and a negative root.

By all means, spend some more time playing around with your algebra tiles, making sure you're really confident how completing the square works.

And remembering that it's another tool to use when you're solving these quadratic equations.

I look forward to seeing you again.

Well done.