In today's video, we're going to cover how to prove that algebraic identities are true. If we use this question as an example, this equation here is known as an algebraic identity, which just means that for any value of x that we pick, this left hand side will always be equal to this right hand side. You can normally tell when an equation is an algebraic identity like this, because instead of having an equal sign in the middle, it will have this symbol instead, which has three lines instead of two, and we call this an identity symbol. Now how to use identity symbols is a whole other topic, but all you need to know for proofs is that if an equation contains an identity symbol, you can't move terms from one side of the equation to the other side. For example in this equation, we couldn't subtract 18 from both sides, to move the 18 term from the left to the right.
That isn't allowed with these identity equations. Instead, in order to show that the identity is true, like we're trying to do in this question, what we need to do is make the left hand side look exactly the same as the right hand side. So here we could keep the right hand side exactly as it is, and just keep rearranging the left side until it looks the same as the right side. In general, the best way to do this is to simplify the left hand side first, by getting rid of any brackets and collecting like terms, and then seeing what we need to do to make it look like the right side. So for our question, we'd first want to expand the 3 sets of x-2 to get 3x-6 plus the 3x and the 18. And then we can combine the like terms to simplify it all to 6x-6.
plus 12. Now this expression here is as simple as we can go, but if we look at it, it's still not the same as our right hand side. If we think about it though, this is because the right hand side has been factorized. So to make our left hand side the same, we need to factorize our 6x plus 12, which we can do by taking out the common factor of 6 and putting the other factors of x plus 2 in the brackets.
And then that's it! We've proved that the identity is true, because we've shown that the left hand side is exactly the same thing as the right hand side. Let's try one more. So for this one, we want to do basically the same thing.
We need to simplify the left hand side first, and then think about how we can rearrange it to make it look like the right hand side. To start, let's expand the n plus 4 squared. And remember, whenever you have a whole bracket being squared like this, it means you have to multiply the whole bracket by itself. So we get n plus 4 times n plus 4, which multiplies out to give us n squared plus 4n plus 4n plus 16. Next, we can expand this other bracket to get minus n squared minus 2n. And then if we simplify all of this, the n squareds cancel out, and we're left with 6n plus 16. Then the last step is to make it look like our right hand side, and for this we need to factorize it by taking out the common factor of 2 to get two sets of 3n plus 8. And that's it, it now looks exactly the same as the right hand side.
One last thing I want to say is that you might sometimes get proof questions like this, that don't contain the identity symbol, and just have an equal sign instead. But as long as they're asking you to prove it, then you still do it in exactly the same way as we've been doing in this video. Anyway, that's it for this topic, so thanks for watching, and we'll see you again soon!