Part a) Factorize 𝑎 squared minus four 𝑏 squared. Part b) Solve two minus three 𝑥 over seven equals eight.
So first of all, we’re gonna take a look at part a. And in part a, we’re asked to factorize. So we’ll remind ourselves what this means. This means putting into brackets. Now we’ve got a special expression here because it’s something called the difference of two squares. And that is the difference of two squares because the first term is a square and the second term is a square. And then we’re subtracting the second term from the first term.
And we’ve got a special relationship for the difference of two squares. And that is if we have our expression in the form 𝑥 squared minus 𝑦 squared, so a square minus another square, then this is equal to 𝑥 plus 𝑦 multiplied by 𝑥 minus 𝑦. So this would be it fully factorized. But why does this work?
Well, if we expanded these brackets, we get 𝑥 multiplied by 𝑥, which is 𝑥 squared, then 𝑥 multiplied by negative 𝑦, which will be negative 𝑥𝑦. Then we’d have positive 𝑦 multiplied by 𝑥, which gives us positive 𝑥𝑦. And then, finally, 𝑦 multiplied by negative 𝑦, which give us negative 𝑦 squared. Well then, negative 𝑥𝑦 and positive 𝑥𝑦 will cancel. So therefore, we’re left with 𝑥 squared minus 𝑦 squared, which is what we started with.
So great, we’ve seen the general rule for the difference between squares and also shown where it’s come from. So now let’s get on and factorize our expression. Well, to work out what goes inside the brackets, we need to find the square root of each of our terms.
Well, first of all, if we find the square root of 𝑎 squared, well, it’s just the inverse of squaring. So the square root of 𝑎 squared is just going to be 𝑎. So therefore, we know that we’re gonna have an 𝑎 at the beginning of each of our brackets. And then the signs inside the brackets, we have one positive, one negative. So we have 𝑎 plus and 𝑎 minus.
And now we need to find out what’s gonna be the second term in each of our brackets. Well, to do that, we need to find the square root of four 𝑏 squared. And we can deal with this in two separate parts, because we can split it up and go what’s the square root of four and what’s the square root of 𝑏 squared. Well, the square root of four is two. And the square root of 𝑏 squared, as we already said, it’s the inverse of squaring. So it’s just gonna leave us with 𝑏. So we get two 𝑏. So therefore, that’s the next term in our brackets.
So then we can say that, fully factorized, 𝑎 squared minus four 𝑏 squared is 𝑎 plus two 𝑏 multiplied by 𝑎 minus two 𝑏. And if we wanted to check it, we could expand the brackets. And when we do that, we get 𝑎 squared. And then we get minus two 𝑎𝑏 plus two 𝑎𝑏 — so they will cancel — and then minus four 𝑏 squared because two multiplied by two is four and 𝑏 multiplied by 𝑏 is squared. Okay, great, so that’s part a. Now let’s move on to part b.
So for part b, there are a number of different ways that you could solve this. I’m gonna demonstrate two just to show two different approaches. The first way would be to add three 𝑥 over seven to both sides of the equation. And that’s because if we’re wanting to solve for 𝑥, I like to always make sure that the 𝑥 term is positive because it makes it easier when we get to the later parts of the question.
So by adding three 𝑥 over seven to both sides of the equation, we’ve got on the left-hand side of the equation negative three 𝑥 over seven. Well, if we add three 𝑥 over seven to this, it’s gonna be canceling it out. So it’s just gonna be zero. So therefore, on the left-hand side of the equation, we’re just gonna have two. And then this is gonna be equal to eight plus three 𝑥 over seven. And that’s because we’ve added three 𝑥 over seven to both sides of the equation. And we have to do that to both sides to keep the equation balanced.
Now we’ve got what we wanted, which is a positive term that involves 𝑥. So now what’s the next step? Well, the next step would be to subtract eight from each side of the equation. And that says that we leave the term involving 𝑥 on its own. We would usually do this is the next step because we’ll deal with anything involving the fraction with the 𝑥 inside afterwards.
So when we do this, we’ll have negative six on the left-hand side. And that’s if you have two take away eight, we get negative six. And then this is equal to three 𝑥 over seven. So now as we want to have our 𝑥 term because we’re wanting to find what 𝑥 is, then we can remove the fraction by multiplying by the denominator. So we’re gonna multiply each side of the equation by seven. So we get negative 42 is equal to three 𝑥.
So great, we’re almost there. We almost found out what 𝑥 is. So now we know what three 𝑥s are, so what do we do to find single 𝑥? Well, again, at each stage, we’ve done the inverse operation. And we’ll do it once more because here we’ve got three multiplied by 𝑥 or 𝑥 multiplied by three. So what I’m gonna do is the inverse and that is divide by three because if we’ve got three 𝑥 as we want to find one 𝑥, we divide by three. And we get one 𝑥, or just 𝑥.
So this gives us our final value of 𝑥, cause we’ve got negative 14 equals 𝑥 or 𝑥 equals negative 14. Okay great, so that’s the first method. I said I’ll show you another method, which I’m gonna do now.
Now for this other method, what I’m gonna do is remove the fraction to begin with. So I’m gonna multiply through by seven because that’s the denominator. But the key thing here is if you’re gonna multiply through by seven, you have to multiply every term by seven. So we’re gonna get 14 — that’s cause two multiplied by seven is 14 — and then minus three 𝑥 — and that’s because if you have three 𝑥 over seven, you multiply it by seven, the denominator cancels out. So therefore, we just have three 𝑥. And then this is equal to 56, cause again we have to multiply by seven, and eight multiplied by seven is 56.
Okay, great, so now what’s the next stage? Well, the next stage will be to add three 𝑥. And we can do that because again, like we did in the previous method, we want to have a positive 𝑥 term. So if we add three 𝑥 to both sides of the equation is gonna do this. So when we do this, we get 14 equals 56 add three 𝑥. We got this because we add three 𝑥 to negative three 𝑥, which gives us zero. And then we have to add three 𝑥 to the other side. So we get 56 plus three 𝑥.
So now what we do is we subtract 56 from each side of the equation. And the reason we’re gonna do that is cause we want the 𝑥 term left on its own. So when we do that, we get negative 42 equals three 𝑥. That’s cause 14 minus 56 is negative 42. And then we’ve removed the 56 from the right-hand side of the equation.
So now all we need to do is find out what 𝑥 is. So to do this, again, it’s an inverse operation. So same as before is- we’re multiplying by three 𝑥, cause it’s three multiplied by 𝑥. So we’re gonna do the opposite or the inverse. So we’re gonna divide each side of the equation by three. So we arrive at negative 14 equals 𝑥, the same as before.
So therefore, we can say that we’ve shown two methods how to solve the problem. And we found that 𝑥 is equal to negative 14. If you were doing it and you’re just using one method, then you might want to check that it’s correct by substituting it back into the equation.