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.

In this lesson, we're gonna be looking at solving quadratic equations using the formula.

We're gonna start by talking a little bit about completing the square.

If you feel like you need a quick recap on that then make sure you do that before proceeding with this lesson.

Our key new concept today is the quadratic formula.

The quadratic formula is a formula for finding the solutions to any quadratic equation.

We are gonna derive this and talk about how to use this in this lesson.

So we're gonna start by deriving the quadratic formula.

What is it, where does it come from, and how does it then help us solve equations? Let's have a look.

So we're gonna start by looking at completing the square.

So completing the square can be used to find solutions to any solvable equation in the form AXE squared plus BX plus C.

We're gonna have a quick reminder of how it works.

So let's try solving 2X squared minus 20X plus 32 equals 0 by completing the square.

So first we can divide through by the coefficient of X squared, so divide everything by two, making sure you divide each term by two.

Zero divided by two is just zero, so that's fine.

Now we need to complete the square.

So X minus five all squared would give us our square.

However, if we do X minus five all squared, that gives us X squared minus 10X plus 25, and we don't want that plus 25.

So we need to subtract 25, and don't forget that we had a 16, so subtract 25 and add 16.

We can rearrange the constants on the right-hand side.

Of course we can add negative 25 and 16 first if we wanted to, it doesn't matter, and then collect any like terms. From here we can square root.

It's really important that we remember that there are two roots, a positive and a negative root.

So you'll see that in front of the square root symbol.

I've put a positive and a negative sign because I want both roots.

Because the square root of nine is three or negative three, I can write that as plus or minus three.

So in order to solve that then I'm going to do three add five, or negative three add five and solve.

So we get three add five, which is eight or negative three add five, which is two.

So Alex says, "I have practise loads of these and I always do exactly the same steps.

Is there a way to generalise this process?" Well actually, Alex, yes there is.

You've got a really good point.

When we're solving by completing the square, we 're literally doing exactly the same thing each time, but with different values.

So what we can do is we can generalise what is happening to come up with a formula where we can just input our values and get our solutions without having to constantly be manipulating that algebra.

So let's start by writing a quadratic equation in the form AXE squared plus BX plus C equals zero.

And that could represent any quadratic equation.

In the previous example, A would've been 2, B would've been negative 20, and C would've been 32.

Remember, this is when an equation is equal to zero.

So we do need to make sure we're in that format first.

What we're gonna look at now is rearranging this general equation to make X the subject.

And the way we're gonna do that is using the completing the square method.

So we're gonna do this step by step.

Feel free to pause the video and think about what the next step would be as we go through.

So, step one.

Remember, we always start by dividing by the coefficient of X squared.

So think about what this would look like if we divided through by A.

Okay, we'd have X squared plus B over A, that's of X, plus C over A.

Zero divided by A will be zero, so we don't need to write zero over A.

Okay, think about what our next step was.

Well, our next step was to half the coefficient of X to work out the expression in the bracket.

So what's gonna happen if we half B over A? So if we half B over A, we can write that as B over 2A.

You think a half multiplied by B over A is B over two multiplied by A, or B over 2A.

Of course then we need to square that, that's the point of completing the square.

And then, remember we need to subtract the square of the constant in the bracket, because when we form that square, we have to do B over 2A all squared.

And then we don't want that constant, so we need to subtract it before adding the constant that we do want.

So that means we're gonna subtract B over 2 A all squared, make sure that's in brackets, and then add the constant that we did want.

So add C over A, and we're still equal to zero.

What we've done there is we've completed the square.

Right, think about what we could do now in step three.

So now we're gonna use a bit of our manipulation skills and we're gonna start by writing B over 2A all squared in a simpler form.

What that looks like then is B squared over 2A squared, and 2A All squared is 4A squared.

So all we've done in that step is just rewritten that B over 2A all squared as B squared over 4A squared.

Step four, what do we do next in our completing the square method? Right, we rearrange the constants on the right-hand side.

We did say you can add them first and then rearrange.

We are going to rearrange first and then we're gonna look at how we're gonna add these.

So we need to add B squared over 4A squared and subtract C over A.

So what we can do now then is we need to combine those constants.

In order to do that, they need to have a common denominator.

Now if you look at the denominator at the moment, they both have a factor of A, we need them both to be 4A squared.

So B squared over 4A squared can stay at as it is, but C over A, if we multiply the numerator and the denominator by 4A, that gives us 4AC over 4A squared.

Just check you're happy with that step before we move on.

Right, step number six, what can we do now to simplify our expression? Well, we can combine the fractions.

That was the point of having a common denominator.

So we could write this as B squared minus 4AC all over 4A squared.

The left hand side of our equation hasn't changed.

Now, think back to our completing the square method.

Once we had a bracket squared equaled a constant, then we could square root.

That right-hand side at the moment is representing a constant.

Remember, X is actually the only variable, A, B, and C are gonna be constants.

So seven, we are going to square root both sides, but we've gotta remember that there's gonna be a positive and a negative root.

For now, I can just write that as plus or minus the square root of B squared minus 4AC all over 4A squared.

Now we're nearly there.

We're just gonna play around with that right-hand side of the equation to make it a little bit simpler.

So what we can do now is we can square root the numerator and the denominator separately just to simplify this a little bit.

Now the numerator, because we've got two values added together, there's not really anything we can do to simplify the square root of that.

So the numerator is gonna stay as plus or minus the square root of B squared minus 4AC.

However the denominator, 4A squared, if we square root that we get 2A.

Right, what do we do next in our completing the square method? We need to make X the subject by adding the additive inverse of the constant that was in the bracket.

So that means subtracting B over 2A.

Can you see something that we could do now? Well, our fractions have a common denominator, so we can simplify by adding.

So we've got X equals plus or minus the square root of B squared minus 4AC.

Subtract B, but make sure that that's not in our square root symbol.

Okay, subtract B all over 2A.

And then look at that, we've got X as the subject.

So we've written a formula for finding X.

To avoid mistakes with a square root, what generally happens is we write it this way round.

So we've just moved the terms around on the top of that fraction.

So we've got negative B plus or minus the square root of B squared minus 4AC all over 2A.

And that is called the quadratic formula.

That's really, really exciting that we have managed to derive that formula for finding the solutions to any quadratic equation using our completing the square knowledge.

There was lots of algebraic manipulation there, it was all things that we have seen before.

So what we've done is we've shown that all quadratics of the form AXE squared plus BX plus C equals 0 can be written in this form.

So if we know what A, B, and C are, we can input it into this formula to find X.

We know that a lot of quadratic equations have two solutions.

So what part of this formula is gonna give us the two solutions? Can you see it? Well done, if you spotted, it's the fact that we've got the positive and the negative square root.

So this is actually gonna be two calculations, one where we're adding the positive square root of B squared minus 4AC, and one where we're adding the negative square root of B squared minus 4AC.

So a quick check of some of the parts of that process.

Which of these is equivalent to half of B over A? How could we write half of B over A? What do you think? Well done if you said B over 2A.

What about this? How could we write B over 2A all squared? Well done if you spotted it was B squared over 4A squared.

Square the numerator and the denominator.

Okay, which of these is equivalent to C over A? Remember, to get an equivalent fraction we need to multiply the top and the bottom by the same value.

So multiplying the top and the bottom by A would give us AC over A squared.

Multiplying the top and the bottom by 4A would give us 4AC over 4A squared.

Fantastic.

Time for you to see if you can replicate that method.

So here are the steps for deriving the quadratic formula.

You will probably notice that they are in the wrong order.

What I want you to do is rearrange them so they are in the correct order and they form our proof of how to get from a quadratic to our solutions.

Give that a go and then we'll run through it.

Well done.

If you haven't written this down already then, this is your opportunity to make sure you've got these in clear steps.

So we should have E, and then A when we divide it by the coefficient.

Then you should have K, which is where we completed the square.

And then B, where we squared the B over 2A.

Then, you should have G where we rearranged, put our constants on the right-hand side.

J and then I, which is where we were adding our fractions by finding a common denominator, and then F where we were square rooting, remembering the plus and minus because we want the positive and the negative square root.

Then D, we simplified our square roots a little bit, and H where we added the additive inverse.

And then C was our final form of our quadratic equation.

Well done if you've got those all in clear steps.

Now we're gonna look at how we can use this.

So using the quadratic formula.

Using the quadratic formula is therefore quite straightforward when you understand where it comes from.

All we need to do is substitute the values for A, B, and C.

So let's look at 2X squared plus 10X minus 28 equals 0.

What's the value of A, what's the value of B, and what's the value of C? What do you reckon? Okay, so A is two, B is 10, and C is negative 28.

Be careful that you include positive or negative values as necessary.

So, check that you are happy with this idea.

In this equation, what is the value of A, B, and C? Have a think and then we'll check our answers.

Lovely, so A is five, B is negative three, and C is one.

Okay, try this one.

Did I catch you out with this one? I've written the terms in a different order.

So A should be the coefficient of X squared, so negative seven.

B is one.

If it's just X, that means 1X.

And C is negative five.

What about this one? What are you gonna need to do first? Have a think.

So I'm gonna need to rearrange.

There's actually options and it depends how you rearrange this.

If you rearranged it as X squared plus 5X minus 3 equals 0, then A would be one, B would be five, and C would be negative three.

If you rearranged it the other way, so subtracted X squared from both sides, you would've got A as negative one, B as negative five, but C as positive three.

They're both gonna give us the same final answer.

So now all we need to do is substitute into our formula.

Remember that we need to put the substituted values in brackets.

So here we would do negative B, so negative 10, 10's our value for B so that's going in a bracket, plus or minus the square root of 10 squared, and again, that's going in the bracket, minus 4 times A times C, so my A and C values are in brackets, including the fact that C is negative 28, all over 2A.

Remember, that plus or minus means we're gonna have two options.

I often evaluate the positive square root first, but it does not matter which way you do it.

And then then negative square root.

So they're the same as above, just one's got the positive square root and one's the negative square root.

And then we just need to evaluate.

If you're doing this without the calculator, you might want to do it in steps.

We get X is two.

And if we try the negative square root, X is negative seven.

So Alex says, "I could have factorised that quadratic.

Why would I use the formula?" Laura agrees, "The formula's gonna be useful when the quadratic does not factorise easily." It's very rare that you are going to be using the quadratic formula without a calculator, because if you can do it without a calculator, you're probably factorising to solve.

There we go, Alex has got it, now we should be able to use our calculators.

Make sure you've got your calculator to hand for the next bit.

So if we've got 3X squared minus 12X plus 4 equals 0, let's start by writing down our values for A, B and C.

So A is three, B is negative 12, and C is four.

You can input the whole formula into your calculator using the fraction button.

Do this at the same time as me.

So we're gonna start by using the fraction button and we've got minus B, haven't we? So we've got minus, and then in brackets negative 12.

Then decide if we're gonna do the positive or the negative square root.

Let's do the positive first.

So we need to do plus, and then we need to type in our square root and then input all our correct values.

So square root of negative 12 all squared, make sure there's a bracket around that, minus 4 times 3 times 4.

Put brackets around those values.

And then use your arrow keys to move to your denominator and you can type in two lots of three.

Then if you execute, you get a value.

Now at the moment, that's probably not in the format we want it, 'cause it says give your answers to one decimal place.

So use a format button, or whatever equivalent button on your calculator, to change the value to a decimal.

We want it to one decimal place, so 3.

6.

Right, Alex says, "Now I need the other solution.

It's gonna take me ages to type that all in again." What could Alex do to efficiently get the other solution? What do you think? Right, all he needs to do is use his arrow keys to go back through his expression and change the positive square root to the negative square root.

So just go back through to where the plus was before your square root and change it to a minus.

And then just press execute and change it to a decimal.

Make sure you round to the right degree of accuracy, so we want 3.

6 and 0.

4.

And that's it.

And we can use that method to solve any quadratic.

I'm gonna run through this again on the left-hand side and then I'd like you to have a go.

So first, identify your A, your B, and your C, and it might be helpful to write that down.

And then I'm gonna get my calculator out and I'm gonna input that quadratic formula.

I've got that written down in front of me so I can use that at the moment.

I'm gonna do one with the positive square root, and one with the negative square root.

Making sure that I am checking what I've put into my calculator.

In your working out, you should write down what you have typed into your calculator.

I know it takes a little bit of time, but it stops you from making any mistakes and it helps you see where you might have gone wrong.

We get 1.

9 or negative 0.

42.

Right, see if you can try this one on your calculator.

Off you go.

So A is negative three, B is negative two, and C is seven.

Inputting into our formula with a positive square root and a negative square root looks like that, and then we just need to trust our calculator skills to get our final answers.

Time for a practise then.

Use your calculators to solve these.

Write your answers to two decimal places.

Off you go.

Well done.

Question two, I'd like you to find the size of the marked angles.

I want your answer to the nearest degrees.

So the nearest whole number of degrees.

Little bit of a clue, vertically opposite angles are equal.

So you need to use that to form an equation before you can solve it.

Off you go.

And finally, Andeep has just learned about the quadratic formula and is using it to solve this equation.

Here are his lines of working out.

I want you to see if you can spot any mistakes he has made and then can you give him a bit of advice? 'Cause I think there's probably a better way for him to do this so he doesn't make those mistakes.

Come back when you're ready for the answers.

Well done.

Pause the video and just check you got these answers.

If you haven't, then check that you've used the quadratic formula correctly and try typing them into your calculator again.

Pay particular attention to the last two, because they need to be equal to zero before you find your values of A, B, and C.

So make sure you've done that.

Okay, question two.

So we can form an equation, 'cause we know these need to be equal to each other.

However, to use the quadratic formula, we need to have our equation equal to zero.

So if we rearrange, so our equation is equal to zero.

Check you've got the same second step as me.

And then we just need to type in our values for A, B, and C into our quadratic formula and we get two answers.

We do need to check if these answers are sensible, and remember the question wanted us to find the size of the angles.

We need to substitute into one of the angles.

It's useful to substitute into both to check you get the same answer.

I'm gonna substitute into 2X squared plus 10X minus 2, and that gives me 146.

I'm gonna use the other answer for X as well, and that gives me an answer of eight to the nearest whole degree.

So the angles are either 146 or eight degrees.

Now both of those are valid answers.

However, if the diagram is roughly to scale, then the angle would be obtuse.

So it would be when X is 6.

5, which gave us 146 degrees.

So just check the question whether it tells you what sort of angle it's supposed to be.

And finally, Andy made a lot of mistakes.

So first, he needs negative nine all squared, so he needs the negative nine to be in a bracket.

Then, he's only done the positive route, he's missed out the negative route.

He should have two solutions.

If we're being really picky, he's over rounded here.

The square root 39 is not six.

So he is lost some accuracy by not keeping his exact answer by rounding it off.

Then he forgot his brackets again, 'cause it should be minus negative nine, with a negative nine in brackets.

So what advice did you give him to improve his accuracy? I wonder if you said something like, "Put brackets around the substituted values." Or, "Input the whole calculation in one go rather than splitting up into steps." Most calculators have a fraction button that you can use, so you can write this in one go.

If you are not using a calculator that will do that, then type in the whole of the numerator and then divide it by 2A.

And then you're gonna complete that in the least number of steps possible.

In this last part of the lesson, we're gonna look at a really cool function that some calculators have in order to help us solve quadratics.

Now, some calculators have a function that uses the quadratic formula for you, all you have to do is put in the values for A, B and C.

Just a word of warning.

It's a really useful tool for checking your answers, but it's still important that you're aware of how completing the square works and how the quadratic formula works.

What that allows you to do is understand the structure behind these solutions.

And we know that having more than one method helps us check for answers.

If you're just typing things into your calculator, you're not showing any working out, you're not showing that you understand what you are doing.

So, let's have a look at this one.

So if your calculator is similar to mine, you want the home button, 'cause you want the different options.

So it's like a home or a menu button, something like that to find the options.

In this calculator, we're gonna use the arrow keys, we want the equation tool.

If you want to select it, if you press OK or if you press the execute button, and this should bring up this screen.

Now, we want to solve a polynomial at the moment.

there's some other exciting things that we can do, for now it's a polynomial that we want to solve.

Now hopefully you notice that the top one is a quadratic expression.

So it's that one that we want.

If it asks you what degree you want, it's degree two.

We want our highest exponent of the variable to be squared.

Okay, so we want that top one that looks like a quadratic, and then it should bring up this screen.

Pause the video, have a play around, see if you can get to this point on your calculator.

Be aware, not all calculators have this function.

And all we need to do now is input A, B, and C.

Make sure you've rearrange your equation to equal zero first.

So type in your value for A and press OK or execute, then it'll move along to B.

Type in your value for B, and do the same for C.

Remember, if your values are negative, you need to type in negative 11, for example.

And press OK or execute again.

Right, the first screen you should see gives you a value for X one.

That's one solution for X.

Of course, if you want it as a decimal, you're to use the format button to change it into a decimal, just like you would normally when you are using your calculator.

Make sure you write down that answer to the degree of accuracy you wish, and then if you press the OK or the execute button again, it should bring up a value for X two.

That's another solution for X.

So you get your second solution, and again, use the format button to change to a decimal.

So now we've got our two solutions.

Just to make you aware of a couple of other bits, if you press OK or execute again, it is likely to give you some other pieces of information about your quadratic.

We don't need those for the moment, but they may be useful later on.

Press OK or execute a couple of times and it should return you to your input screen.

What that means is if you think you've typed something in a little bit wrong, you can change those and press execute again.

So just check that you've typed in A, B, and C correctly, especially if your answer wasn't what you were expecting.

So, Lucas is gonna give this a go.

So these are the screens on his calculator.

He gets a strange answer.

He thinks he has gone wrong.

What has he done wrong? Can you work it out? Well done if you spotted he has not rearranged to equal zero.

A should be five, B should be negative nine, and C should be negative five.

Or, if you rearranged it the other way, A is negative five, B is nine, C is five.

Izzy is checking her answer.

She solved this already and got the answer X equals 1.

5, X equals one, but she's checking her answer.

Have a look at what she's typed in.

She does not get the same answer.

Is her calculator correct or do you think she was correct to start with? What do you reckon? If you look closely, her B value was negative five.

She typed in positive five.

Sometimes on these calculators it's really hard to see those negative symbols, so just take care that you are typing in the correct value and double check before you execute.

It's really good that Izzy took the moment to double check her calculator.

We don't always trust our calculators, guys, 'cause they're only as good as the values that we've typed into them.

It's easy for us to make mistakes typing into a calculator, so do not trust it blindly.

Time for you to have a go.

So Izzy and Lucas have had a go at doing a solving equations homework.

They are helping each other out by checking their solutions.

These are Izzy's answers.

I'd like you to help Lucas check which ones are incorrect by using your calculator.

Let's have a look then.

First one was correct.

The second one, the first solution was correct, but the other solution was incorrect.

It might not necessarily have been that way round on your calculator, it doesn't matter which order it brings up the solutions.

C was correct.

D, let's check our value for A, B, and C, so we need to rearrange to equal zero.

So A is negative one, B is nine, C is negative eight, and that one was correct.

For E, A was two-thirds.

That's okay, you can type in using the fraction button two over three and it will input that as a coefficient of X squared.

That's absolutely fine.

And lastly, we need to rearrange.

I've decided to add a half X squared so that I have positive coefficient of X squared.

So I've got half X squared plus 2X minus 7 equals 0, and then that one was incorrect.

Fantastic, we're now experts at using the quadratic formula and hopefully we're getting really confident with using our calculators.

If we can use our calculators quickly and efficiently, that's gonna save us time when we're looking at more complicated questions.

Would like to remind you, it's really important that you write your working out down as well so that somebody else could follow along the process you were doing.

That's what mathematics is all about, is writing things so other people can read it and follow it.

It's like providing an argument for why we are right.

Fantastic, have a read through what we've learned today, and I hope you can then put that into practise when you're solving quadratic equations in the future.

Thank you for joining us, it'd be lovely to see you again.