Video: AQA GCSE Mathematics Higher Tier Pack 5 • Paper 1 • Question 25

Expand and simplify (2𝑥 − 6)(3𝑥 − 2)(3𝑥 + 2).

05:52

Video Transcript

Expand and simplify two 𝑥 minus six multiplied by three 𝑥 minus two multiplied by three 𝑥 plus two.

So in this question, what we need to do is expand three brackets. Well, the way that we do it is to actually expand two brackets first and then multiply everything inside the answer from expanding two brackets by the other bracket. So what we’re gonna start with in this question is three 𝑥 minus two multiplied by three 𝑥 plus two.

So the reason I’m gonna expand these two first is because they are the difference of two squares. And what this means is that they give a known answer or a known response when we expand the brackets. So I’m gonna expand them in the normal way. But we’re gonna see what answer we should get. So the result we’re expecting is well if we’ve got 𝑎 minus 𝑏 multiplied by 𝑎 plus 𝑏, so in this case three 𝑥 minus two multiplied by three 𝑥 plus two where our 𝑎 is three 𝑥 and our 𝑏 is two, then the result we should expect is 𝑎 squared minus 𝑏 squared. So in our case, that will be three 𝑥 all squared minus two all squared.

So if we’re gonna expand three 𝑥 minus two multiplied by three 𝑥 plus two, first of all, we need to multiply the three 𝑥 in the first bracket by both terms in the second bracket. So we get three 𝑥 multiplied by three 𝑥 which is nine 𝑥 squared. And then, we’ve got three 𝑥 multiplied by positive two which gives us positive six 𝑥. So we can see now that we’ve multiplied our three 𝑥 by both of the terms in the second bracket.

So now we need to multiply the negative two from the first bracket by both of the terms in the second bracket. So first of all, we have negative two multiplied by three 𝑥 which gives us negative six 𝑥 and then negative two multiplied by positive two which gives us negative four. So we’ve got nine 𝑥 squared plus six 𝑥 minus six 𝑥 minus four. Well, now if we simplify, we can see we’ve got plus six 𝑥 minus six 𝑥. So these will cancel, which gives us nine 𝑥 squared minus four.

Now is this the result we were expecting? Well, we were expecting 𝑎 squared minus 𝑏 squared, so in our case three 𝑥 squared minus two squared. Well, if you square three 𝑥, you get nine 𝑥 squared and if you square two, you get four. So this is the result we’re expecting. Okay, so now we can rewrite our expression. So now, we have two 𝑥 minus six multiplied by nine 𝑥 squared minus four.

So what we’re gonna do now is expand these two brackets. So we can use the same method as before. So we’re gonna multiply the first term in the first bracket by the first term in the second bracket. So we got two 𝑥 multiplied by nine 𝑥 squared which is gonna give us 18𝑥 cubed. And just reminding ourselves how we actually did that, if you’re multiplying two 𝑥 by nine 𝑥 squared, first of all, you multiply the coefficients, so two multiplied by nine. That gives us 18. Then, you’ve got 𝑥 multiplied by 𝑥 squared. So you add the powers. So we’re gonna get 𝑥 cubed.

Then next, we multiply the two 𝑥 from the first bracket by the negative four in the second bracket. So that gives us negative eight 𝑥. Then, we’ve got the second term in the first bracket which is negative six multiplied by the first term in the second bracket. So that’s gonna be negative six multiplied by nine 𝑥 squared which gives us negative 54𝑥 squared. And then, finally, we’ve got plus 24. That’s cause you got negative six multiplied by negative four. Negative multiplied by a negative is a positive. So we got plus 24.

So what we’re gonna do now is simplify. And to simplify, we can’t collect any like terms because we don’t have any. And that’s because we’ve got different powers of 𝑥 in each term. And you can’t add or subtract if you got different powers of 𝑥. But what we can do is we can rearrange it in descending powers of 𝑥. So therefore, we can say that if we expand and simplify two 𝑥 minus six multiplied by three 𝑥 minus two multiplied by three 𝑥 plus two, we get 18𝑥 cubed minus 54𝑥 squared minus eight 𝑥 plus 24.

So what I want to do now is check our answer. And I’m going to do that so that I can show you another method for expanding brackets that you might prefer to use. So we’re gonna check our answer by expanding two 𝑥 minus six multiplied by nine 𝑥 squared minus four. And the method I’m gonna show you is this one here, which is a grid method for expanding brackets.

So as you can see, I’ve got two 𝑥 and then minus six or negative six along the top. And then, I’ve got nine 𝑥 squared and negative four down the left-hand side. So first of all, I multiply the nine 𝑥 squared and the two 𝑥. So it gives us 18𝑥 cubed. And then, I multiply the negative six and the nine 𝑥 squared which gives us negative 54𝑥 squared. Then, I’ve got the two 𝑥 and the negative four, so that’s negative eight 𝑥, and then finally negative six and negative four which gives us our 24.

And then, all we do is we write out the terms in the grid. So we’ve got 18𝑥 cubed minus 54𝑥 squared minus eight 𝑥 plus 24. So that’s giving us the same answer that we got using the other method. So we can say that we are happy that 18𝑥 cubed minus 54𝑥 squared minus eight 𝑥 plus 24 is the correct expansion.

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