Let's look at how to solve quadratic equations using the Quadratic Formula. We also factorise quadratic expressions and use iteration to find approximate solutions. Quadratic algebraic equations contain terms up to x^2; the highest power for quadratic equations is 2. They are polynomial equations because they consist of two or more algebraic terms.
At GCSE the solutions to polynomial equations such as quadratics will always give real numbers but they can be either irrational and rational numbers.
This video is perfect for AQA GCSE Maths students studying Module 2: Algebra and anyone who simply wants to learn more about solving quadratic equations.
We are looking at solving quadratic equations and we're going to focus mainly on three methods, this lesson. So we're going to look at quadratic formula, completing square and factor rising, if we have time to do, will look at iteration, but it's a bit of a beast. So it's almost an entire video in itself to look at that. So let's look at the quadratic formula first. So what is it?
It's basically the laziest of the three methods. So what we do is we take our quadratic formula, which is the thing in this box, and we plug our values into it. So, for example, if we look at this example question, we're solving for X, we've got three X squared plus five, x plus six equals zero. And so going by this standard layout of axe squared plus BX plus C, in this case, the number in front of X squared is three. So A is three, the number in front of X is five.
So B is going to be five and the number on its own, the constant, is going to be six. So C is six. And all we would do is we would put that into our formula. So we would say X is equal to minus five plus or minus the square root of B squared, which is going to be five squared minus four times A, which is three times C, which is six. And that's all divided by two A.
So two times three. And we can plug that into our calculator. One of the versions of this, we would use the plus version of the plus minus sign and we would then go back and use the minus version of the minus sign. So we should be getting two solutions. We normally expect to get two solutions, which makes sense, right?
Because if we look at the standard shape of a quadratic, it looks something like this. And so when we're solving for this quadratic equaling zero, what we expect, because it crosses zero twice, it crosses it there and it crosses it there if this is our Y and our X axis. So that is the quadratic formula. Next up, we are going to look at completing square. This may be the more complex of the three.
So there's a general rule which we're going to kind of derive as we go.
Again, let's start off with an example. So let's say we've got X squared plus six, x plus five equals zero, and we want to complete the square on this. Well, what we can do is we kind of focus on the first two terms, right? We focus on the X squared and the external. We almost kind of ignore the plus by the constant B at the end.
And what we do is we transform this into something that looks like this, and we take the number in front of X and we have it and we ignore the X squared. So X squared becomes x, and this number, in this case, six halves to become three. And then we square that and you'll see why we do this in a second. And then obviously we've got that plus five equals zero. Well, why do we do this?
What's the point? Well, let's look at what happens if we actually expanded these brackets. So if we take this x plus three squared and we actually expand it, well, this is just x plus three times x plus three, which might make it slightly easier for anyone following along to understand what I'm doing next. So we're going to end up with x times x to get X squared. We're going to end up with x times three to get three x.
We're going to end up with three times x to get three x. And finally we're going to do three times three to get nine. But we know that three x plus three x is going to be six x. And so this is just going to be x word plus six x plus nine. And again, if we were to finish off the rest of the equation, plus five equals zero, plus five equals zero, and we can probably start seeing now the similarities that occur.
So what we end up with is we end up basically recreating the original equation, right? So if you look at what we've got in the original equation, we've got an X squared, we've ended up with an X squared. And also in our original equation, we've got a plus six x, we've got a plus six x. So they're both correct. And obviously we've got the plus five and plus five, we've got this pesky plus nine hanging around.
And we don't want that, right? We don't want this plus nine to be there because what we can see is at the moment, these two versions of the equation aren't the same as each other because we've got this extra plus nine hanging around. So what we see we need to do is we need to go back to our completed the square version and we need to subtract nine from it. And therefore we'll end up subtracting nine here and subtracting nine here. And we can then simplify this because those nine will then cancel and we end up with x squared plus six x plus five equals zero.
So what we can see is this light.
This new equation is the completed square version of the above original quadratic equation. And we can simplify this slightly, right, because we know minus nine plus five is minus four. And so we would finalise this by saying x plus three squared minus four equals zero. And this would be our completed square version. And we can now rearrange for x, can't we?
Because we know x plus three squared, we can add four to both sides. So we can get x plus three square plus four and we can then square root both sides. And so x plus three equals plus or minus square root of four, which is just going to be plus or minus T.
And finally, we can subtract three from both sides. So x equals minus three plus or minus two. And that would be our answer. So that's a brief intro to complete square. And why we do it right is because the thing to always remember is we end up with this pesky extra term coming through and so we always attract that to avoid changing the whole value of the left hand side of the equation.
Moving on to the final version factorising. So I've done a simple example here. All we need to do here is we want to get it into the format of two brackets, like that equals zero. And I know that to get x squared, I've got to have x here and x here. And to get five x, well, I know that whatever I put here in this box and whatever I put here in this box, I'm going to multiply x by this box and I'm going to multiply this x by this box.
And so I'm going to end up with something x plus something x equals five x. So I've got to think of two numbers that adds together to get five. I also know that I'm going to have to multiply the two boxes together, right? I'm going to have one box times another box, and that's going to give me my new number that's going to give me six. So I've got to think of two numbers at times together to get six and as together to get five.
Well, I know there's two numbers, they're going to be three and two. And so that's how we can use factorising to again solve quadratic equations, right? Because I know that therefore, if I just focus on the left bracket and I decided to make the left bracket equal to zero, well, therefore the whole left hand side equals zero because zero times anything is just zero. And so what does x have to be equal for this bracket to be zero? Well, something plus three equals zero.
So I know they're thought that something has got to be minus three, right? Because minus three plus three equals zero. If I look at the right hand bracket and I focus on making this equal to zero, I know that something plus T equals zero. And therefore x has to be minus t because minus t plus t gives me zero. So that's a brief intro to all of our three main topics on solving critical reasons.
Let's go and look at one example. So let's do this with the quadratic formula. So in this case, I'm going to imagine there's basically a one in front of this x squared, right? It's the same as multiplying it by one. So in this case, A is going to be one, B is going to be ten, and C is going to be 24.
And so I do minus ten plus or minus the square root of ten squared, 100 minus four times A times C, which is going to be 96, all divided by two A. So two times one, well that just gives me minus ten plus or minus the square root of four. Square root of four is T, and that's divided by t. Well that means that I'm either going to have minus ten plus t to get minus eight minus eight over t gives me minus four, or I'm going to have minus ten minus t. Well, minus ten minus two is minus twelve, and minus twelve divided by t gives me minus six.
So there are my two values for X. We could do the same thing again, but we could complete square, right? So let's do the same thing, the same question, but we'll do completing the square instead. So again, I'm going to have my brackets like this with my square here, and instead of X squared, I have X and I have this number. So instead of ten, I have five.
And I know that when I expand these brackets out, I'm going to get an extra term hanging around, which is going to be five times five. So I've got to subtract that. And that means therefore, that this whole thing now represents the first two cent. And observing when I have mine plus 24 on the end equal zero, well, I know that one. So I end up with X plus five squared minus one equal zero, and therefore x plus five squared equals one.
If I just add one to both sides. If I square root both sides, I get X plus five equals plus or minus the square root of one is just one. So it's just plus or minus one. And I then get minus five from both sides, I get X equals minus five plus or minus one. So obviously minus five plus one gives me minus four, minus five, minus one gives me minus six.
So again, we've got the exact same two solutions using a completely different method. Finally, let's do method three, right? So again we've got X squared plus ten, x plus 24. Well, in this case, I want two brackets bracket here and I'm going to set that equal to zero. Well, I've got to think of two numbers, right?
I'm going to have X and X here. I've got two numbers that adds together to get ten and times together to get 24. Well, I know those two numbers are going to be six and four, six plus four is ten, six times four is 24. And so if I focus on this first bracket, making this first bracket equal to zero, I know that X has to be minus six. If I then focus on the second bracket, and I want to make this second bracket equal to zero, I know X has to be minus four.
And so again, we end up with the exact same two solutions. And that's where we can apply any one of those three methods to solving our quadratic equations.
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