Find the solution set of five 𝑦 squared plus 24 𝑦 minus five equals zero in the set of real numbers.
So that means we’re looking for real values of 𝑦 which satisfy the equation five 𝑦 squared plus 24 𝑦 minus five is equal to zero. Now I’m going to factorise this quadratic expression here and put it equal to zero. And then using that factorised or factored version of the expression, I’ll be able to quickly identify what the solutions are.
Now this isn’t a completely trivial exercise to factorise this because we’ve got five 𝑦 squared. So we have to think carefully about what numbers we’re gonna put into these brackets. But there’s a nice little technique which can actually help us to do that more quickly. So we take the coefficient of 𝑦 squared, and we take the constant term, and we multiply them together. So in this case, that’s five times negative five which is negative 25. Now we need to list out all the factors of 25. So that’s one and 25 or five and five. But remember, actually, we’re trying to generate an answer of negative 25. So one of those is gotta be positive and one of those gotta be negative.
Now when we’re deciding which of these factor pairs to use, we’re gonna consider the coefficient of the 𝑦 term in the middle of that expression there, so positive 24. So given that one of those numbers in our factor pair is positive and one is negative, when I add them together, I want to get this answer of positive 24. So looking at these, I mean basically if one of those is positive and one of those is negative, when I add them together, I’m gonna get an answer of zero. So that’s no good. In this case, if I make the one negative and the 25 positive, when I add them together, I get positive 24. So that’s the factor pair I’m going to use.
Now the way that I’m gonna use that factor pair is to reexpress positive 24 𝑦 as either negative one 𝑦 plus 25 or as 25 𝑦 take away one 𝑦. So hopefully, you can see that positive 25 𝑦 take away one 𝑦 is still 24 𝑦. So this line here is entirely equivalent to the line above. Now it also doesn’t matter whether I put plus 25 𝑦 minus one 𝑦 or whether I do minus one 𝑦 plus 25 𝑦. But, my next step is gonna be to factorise the first half of that quadratic expression and then to factorise the second half of the quadratic expression. And it just turns out to be slightly easier if I choose the term here that’s got the most in common with that 𝑦 squared term here. Now this has got- I’ve got a 𝑦 in common and I’ve also got a factor of five in common for those too. As I say, it doesn’t actually make any difference to the final answer, it just makes it, I think, slightly easier if we do it this way round.
So factoring that first off, that five is a common factor and 𝑦 is a common factor. So what do I need to multiply five 𝑦 by in order to get five 𝑦 squared plus 25 𝑦? Well, in the first case, I’m just gonna multiply by 𝑦, five 𝑦 times 𝑦 is five 𝑦 squared. What do I need to multiply five 𝑦 by to get positive 25 𝑦? Well, that’s gonna be positive five. So five 𝑦 times 𝑦 is five 𝑦 squared, that’s good. And five 𝑦 times positive five is positive 25 𝑦, so that’s correct.
Now I’m gonna take the contents of that bracket and I’m gonna repeat it over here. And before I do that, it’s important to check that I fully factored that expression. So for example, if I’d have only taken five out as a common factor, then that’s gonna mess up the rest of the method. Or if I had only taken the 𝑦 out as a common factor, that would’ve also messed up the rest of the method. And now I need to think, what do I need to multiply 𝑦 plus five by to get this thing up here, negative one 𝑦 take away five. Well, I need to multiply it by negative one because negative one times 𝑦 is negative one 𝑦, and negative one times positive five is negative five.
So now we’ve written yet another version of this quadratic equation. So five 𝑦 squared, as we said at the beginning, plus 24 𝑦 minus five is equal to zero is the same as saying five 𝑦 squared plus 25 𝑦 minus one 𝑦 minus five is equal to zero. And that’s the same as saying that five 𝑦 times 𝑦 plus five minus one times 𝑦 plus five is equal to zero. And you’ll see that I’ve got this 𝑦 plus five in common with those two expressions. So I’m gonna take that out as a common factor. So I’ve got 𝑦 plus five; in the first case, I’m multiplying that by five 𝑦 and in the second case, I’m multiplying it by negative one. So I could rewrite that whole expression as 𝑦 plus five times five 𝑦 minus one is equal to zero.
There, we factorised the quadratic expression. Let’s just quickly check it before we move on. So 𝑦 times five 𝑦 is equal to five 𝑦 squared, 𝑦 times negative one is negative 𝑦, five times five 𝑦 is plus 25 𝑦, and negative five- sorry positive five times negative one is negative five. And when I simplify all of that, I do indeed get five 𝑦 squared plus 24 𝑦 minus five which is, you know, the expression that I was looking for at the beginning.
So now, we’ve got that factorisation done; we can go on and solve the question. So 𝑦 plus five times five 𝑦 minus one is equal to zero. We’ve got two things multiplied together to make zero. Well, the only way that that can happen is if one of those things is zero. So either the 𝑦 plus five must be equal to zero or the five 𝑦 minus one must be equal to zero. Well, if 𝑦 plus five is equal to zero, then 𝑦 must be equal to negative five. And if five 𝑦 minus one equals zero, then we can add one to both sides to get five 𝑦 equals one, and then divide both sides by five to get 𝑦 equals a fifth.
So our two real answers are: 𝑦 equals negative five or 𝑦 equals a fifth. And to put those answers in a solution set, we’ve got the answer this set of numbers is negative five and one-fifth.