CA Algebra I: Quadratic equation

or on problem 53 it says Tony is solving this equation by completing the square ax squared plus BX plus C is equal to 0 where a is greater than zero so this is just a traditional quadratic right here and let's see what they did let's see first he subtracted C from both sides and he got ax squared plus BX is equal to minus C okay that's fair enough and then let's see let's see divided both sides by a right that's fair enough he got minus here a which step should be step three in the solution so he's completing the square so essentially he wants this to become a perfect square so let's see how we can do that so we have x squared plus B over a X is an unbelievable space here is equal to minus C over a so for this to be a perfect square we have to add something here we have to add a number and we learned from several videos in the past and we kind of pseudo proved didn't actually have several videos right I do it solely on completing the square you sense you have to add whatever it number this is add half of it squared and if that doesn't make sense you watch the Khan Academy video on completing the square but what is half of B over a well it's B over 2a right now what's be over so one-half times B over a is equal to B over 2a and then we want to add this squared so let's add that to both sides of this equation so we're left with x squared plus B over a X and we want to add this squared plus B over 2a squared is equal to minus C over a well do you think you add to one side of the equation you have to add to the other so we have to add that to both sides plus B over 2a squared and let's see with if we've solved the problem so far what they want X B over two right this is exactly what we did x squared plus B over a plus B over 2a squared and they add it to both sides equation so D is the right answer now if you find that little confusing or if it wasn't intuitive for you I don't want you to memorize the steps watch the Khan Academy video on completing the square next problem 56 no 54 alright this is another one that should be cut and pasted alright four steps to derive the quadratic formula shown below I said in previous videos that you can derive the quadratic formula by completing the square and we actually do that in another video I don't want to give too much of a plug for other videos but let's see what they want to do what is the correct order of these steps so the first thing you want to start off with is just like a quad is just a quadratic equation and this one is the first step right this is where we started off with in the last problem then what you want to do is add half of this squared to both sides so B over 2a squared you want to add to both sides and that's what they did here so our order is 1 and then you want to do 4 right that's what we did in the last problem we did 4 and then from here you know that this expression right here is going to be equal to X plus B over 2a squared and once again watch the completing the squared video if that didn't make sense but the whole reason why you added this here so that you know that ok what two numbers when I multiply them equal B over 2a squared and when I add them equal B over a well that's obviously B over 2a all right if you hit twice you're gonna get B over a if you squared you're gonna get this whole expression so you say oh this is just X plus B over 2a squared and you get that there and then is equal to and then they just simplify this fraction they found a common denominator and all the rest and so the next step is step two and then all you have left is step three and you've pretty much derived the quadratic equation so 1 4 2 3 1 4 2 3 that's choice a problem 55 which of the solutions ok so then I'll write I'll put all the choices down so which is one of the solutions to the equation so immediately when you see all of the choices they have these square roots and all of that this isn't something that you would factor you would use a quadratic equation here so let's do that so the quadratic equation is so this is ax squared ax squared plus BX plus C is equal to 0 the quadratic equation is minus B or they do it lower case plus or minus the square root of b squared minus 4ac all of that over 2a and this is just derived from completing the square with this but we do that in another video and so let's substitute it in what is B B is minus 1 right so minus minus 1 that's a positive 1 plus or minus the square root of b squared minus 1 squared is 1 minus 4 times a a is 2 times 2 times C C is minus 4 so x minus 4 minus 4 all of that over 2a a is 2 so 2 times a is 4 so that becomes 1 1 plus or minus the square root so 1 so minus 4 times a 2 times a minus 4 so those that's the same thing as a plus 4 times 2 times a plus 4 right let's just take that minus out so it's plus and then there's no minus here right so let's see 4 times 2 is 8 times 4 is 32 plus 1 is 33 33 all of that over 4 see we're not quite there yet well they say which is one of the solutions to the equation so let's see if we wanted to simplify this out of it oh this was right here right because we have 1 plus or minus the square root of 33 over 4 well they wrote just one of them they wrote just the plus so C is one of the solutions the other one would have been if you had a minus sign here anyway next problem 56 56 this is another one need to cut and paste says which statement best explains why there is no real solution to the quadratic equation okay so I already have a a guess of why this won't have a solution but in general you know let's try the quadratic equation before even looking at the problem let's get an intuition it's negative B plus or minus the square root of b squared minus 4a see all of that over 2a my question is to use what when does this not make any sense well you know this will work for any be any 2a but when does a square root sign really fall apart at least when we're dealing with real numbers and that's a clue is well it's when when you have a negative number under here right if this if you end up with a negative number under the square root sign at least if we haven't learned imaginary numbers yet you don't know what to do you you you there's no real solution to the quadratic equation so if b squared minus 4ac is less than zero you're in trouble there is no real solution right you can't take a square root of a negative sign if you're dealing with real numbers so that's probably going to be the problem here so let's see what b squared minus 4ac is you have B is 1 so 1 minus 4 times a a is 2 2 times C is 7 and sure enough 1 times 4 times 2 times 7 is going to be less than 0 so let's let's see what they have here write the value of 1 squared all right it's b squared is y a well 1 squared same thing as 1 1 squared minus 4 times 2 times 7 sure enough is negative so that's why we don't have a real solution to this equation next problem I'm actually out of space let me alright ok they want to know the solution set to this quadratic equation I'll just copy and paste the so that's essentially the the set of the X's that satisfy this equation they don't want and obviously if for any X that you put in this the left-hand side is going to be equal to 0 so what X's are valid and they just want us to apply the quadratic equation so we've written it a couple of times but let's just do it straight up so it's negative B B is 2 right so it's negative 2 plus or minus the square root of b squared well that's 2 squared minus 4 times a a is 8 times C which is 1 all of that over 2 times a so 2 times 8 which is equal to minus 2 plus or minus the square root of 4 4 let's see 4 I write this down negative B negative B plus or minus the square root of B squared B squared minus 4 times a times C right so you get 4 minus 32 that's why I was double checking to see if I did this right because I'm gonna get a negative number here all of that over 16 and so we're gonna end up with the same conundrum we had in the last I mean at 4-30 two we're gonna end up with minus two plus or minus the square root of minus 28 over 16 and if we're dealing with real numbers I mean there's there's no real solution here and at first I was worried I thought I made a careless mistake or there was an error in the problem but and I look at the choices and I hope they have choice D and all a copy and paste choice D here choice D no real solution so that's the answer because you can't take a square root of a negative number and stay in the real number and they're in the set of real numbers let's see do have time for another one I'm over the 10 minutes I'll wait for the next video see you soon