Solving quadratic equations — Harder example

- [Instructor] What are all the solutions to the equation above? And we have x plus three times x minus five is equal to five. You wanna be very careful here because you're probably have some experience with algebra that, hey, once I factored it out, maybe I could say, okay, maybe this needs to be equal to five, or this needs to be equal to five. That is not the way that it works. That would not actually logical sense. In order to kind of make the factoring useful, you have to be able to say, hey, the product of these two things is equal to zero, because if the product of two things is equal to zero, then you know that either one or the both of them need to be equal to zero. So we've actually have to do a lot of algebraic manipulation here to get it into that form, but let's see if we can do it. So the first thing I would do is just multiply out x plus three times x minus five. So what's that going to be? Well it's going to be x times x, which is x squared, plus x times negative five. So I could say that's minus five x. Plus three times x. So that's plus three x. Plus three times negative five. So that's minus 15. And that's all going to be equal to, and that's all going to be equal to five. And let's see what we can do from here. We could say this is gonna be x squared and then these two terms, right over here, we can add together. Negative five x plus three x is negative two x. And then we have minus 15 is equal to five, is equal to five. Now let's see. We could now subtract five from both sides. So you subtract five there. You subtract five there. And we're starting to get to the home stretch. And we would get x squared minus two x minus 20 is equal to zero. So now we're in business. Now we have this quadratic in a form where we just need to figure out what x values are gonna make this expression equal to zero. And the first temptation is see well can we, is there, is there any way that we can naturally factor this? So is there a, let's see. Are there any two factors of 20 that if I, and it's going to be, one of them's gonna be positive and one's going to be negative since their product gets us negative, where I add 'em together I get to negative two. So let's see. Nah, nothing jumps out. You have four and five, two and ten. Yeah, nothin's jumpin' out at me. So we can just resort to the quadratic formula here. When the quadratic formula tells us that if I have ax squared plus bx plus c is equal to zero, then the solutions of this quadratic equation are going to be x is equal to negative b plus or minus the square root of b squared minus four ac all of that over two a. And I don't tell people to memorize a lot in life, but the quadratic formula is one of those things that it's not a bad idea to memorize. But you should also watch the Khan Academy videos on how this proves and this comes just naturally out of completing the square on this thing right over here. So also understand what it's actually saying but it's also a good thing to know, because just like that we can apply it to this. So in this context, our a is one. It's the coefficient out here that's implicitly out there. So a is one. b is negative two. b is equal to negative two. That's this coefficient right over there. And c is negative 20. c is equal to negative 20. So the roots are going to be x is equal to negative b. So it's gonna be negative of negative two. So negative of negative two is gonna be positive two, plus or minus the square root of b squared, which is four, minus four times a, which is one, times negative 20. So instead I'll put a 20 here. And since that's a negative 20 but I'm subtracting it, I could put a plus there. And then all of that over two a. Well a is just one. All of that over two. So it's gonna be two plus or minus the square root of four plus four times 20 is 80, of 84 over two. Now already, if I'm under time pressure, I already see some choices that are starting to look a little bit like this. But let's see if we can get to the right solution here. So can I simplify this more? Well see the square root of 84. 84, that's going to be divisible by 4. 84 is the same thing as four times 21, and all of that over two. And this is going to be the same thing as, I'll scroll down a little bit, get a little bit more space. This is going to, x is going to be equal to two plus or minus. So this would be the same thing as the square root of four times the square root of 21, which of course is two times the square root of 21, all of that over two. So if you divide each of these by two, which we are doing right here, it's going to be one plus or minus the square root of 21, which is this choice right over there. So it looked like a fairly benign thing, but we had to multiply it out, set it up in kind of a form where the quadratic formula would apply, and we got a fairly hairy answer.