Solving quadratic equations — Harder example

what are all the solutions to the equation above and we have X plus 3 times X minus 5 is equal to 5 you want to be very careful here because you probably have some experience with algebra that hey once I factored it out maybe I could say ok maybe this needs to be equal to 5 or this needs to be equal to 5 that is not the way that it works that would not make actually logical sense in order to in order to kind of make the factoring useful you have to be able to say the product of these two things is equal to 0 because if the product of two things is equal to 0 then you know that either one or both of them need to be equal to 0 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 3 times X minus 5 so what's that going to be well it's going to be x times X which is x squared plus x times negative 5 so I could say that's minus 5x plus 3 times X so that's plus 3x plus 3 times negative 5 so it's minus 15 and that's all going to be equal to that's all going to be equal to 5 and let's see what we can do from here we could say this is going to be x squared and then these two terms right over here we can add together negative 5x plus 3x is negative 2x and then we have minus 15 is equal to 5 is equal to 5 and let's see we can now subtract 5 from both sides so you subtract 5 there you subtract 5 there and we're starting to get to the homestretch and we would get x squared minus 2x minus 20 is equal to 0 so now we're in business now we have this quadratic in a form where we used to need to figure out what x value is going to make this expression equal to 0 and the first is a temptation to 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 2 factors of of 20 that if I and it's going to be 1 is going to be someone's going to be negative since their product is gets us negative or if I add them together I get two negative two so let's see no no nothing jumps out you have four and five two and ten yeah nothing's jumping out at me so we can just resort to the quadratic formula here when the quadratic formula tells us that if I have a x 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 4ac all of that over 2 all of that over 2a 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 proved 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 1 it's the coefficient out here that's implicitly out there so a is 1 B is negative 2 B is equal to negative 2 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 was going to be negative of negative 2 so negative of negative 2 is going to be positive 2 plus or minus the square root of B squared which is 4 minus 4 times a which is 1 times negative 20 so instead I'll put a 20 here and sits up so negative 20 but I'm subtracting it I could put a plus there and then all of that over 2a well a is just 1 all of that over 2 so it's going to be 2 plus or minus the square root of 4 plus 4 times 20 is 80 of 84 over over 2 now already if I'm under time pressure I already you know see some choices that are starting to look a little bit like this but let's see if we can if we can if we can 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 that's going to be divisible by 4 84 is whoops 84 is the same thing as 4 times 21 and all that over 2 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 2 plus or minus so this would be the same thing as the square root of 4 times the square root of 21 which of course is 2 times the square root of 21 all of that over 2 so if you divide each of these by 2 which we are doing right here it's going to be 1 plus or minus the square root of 21 which is which is this choice right over there so it looked like a fairly benign thing but we had to we had to multiply it out set it up in kind of a form where the quadratic formula would a perform formula would apply and we got a fairly fairly hairy answer