Solving quadratics by taking square roots: challenge

in this video I'm going to do several examples of quadratic equations that are really of a special form it's really a bit of warm-up for the next video that we're going to do on completing the square so let me show you what I'm talking about so let's say I have 4x plus 1 squared minus 8 is equal to 0 now based on everything we've done so far you might be tempted to multiply this out then subtract 8 from the constant you get out here and then try to factor it and then you're going to have you know X minus something times X minus times X minus something else is equal to 0 when you're say oh one of these must be equal to 0 so you're going to you know X could be that or that we're not going to do that this time because you might see something interesting here we can solve this without factoring it and how do we do that well what happens if we add 8 to both sides of this equation then the left-hand side of the equation becomes 4 X plus 1 squared these 8's cancel out the right-hand side becomes just a positive 8 now what can we do to both sides of this equation and this is just kind of straight vanilla equation solving this isn't any kind of fancy factoring we can take the square roots of we could take the square root of both sides of this equation we could take the square root so 4 X 4 X plus 1 I'm just taking the square root of both sides you take the square root of both sides and of course you want to take the positive and the negative square root because 4x plus 1 could be the positive square root of 8 or it could be the negative square root of 8 so 4x plus 1 is equal to the positive or negative square root of 8 instead of 8 let me write 8 as 4 times 2 we all know that's what 8 is and obviously the square root of 4x plus 1 squared is 4x plus 1 so we get 4x plus 1 is equal to we can factor out the 4 or the square root of 4 which is 2 is equal to the plus or minus times two times the square root of two right square root of 4 times square root of 2 is the same thing as square root of 4 times the square root of 2 plus or minus square root of 4 is that 2 right there now it might look like a really bizarro equation with this plus or minus two times the square root of 2 but it really isn't this is just these are actually two numbers here and we're actually simultaneously solving two equations we could write this as 4 X plus 1 is equal to the positive 2 square root of 2 or 4 X plus 1 is equal to the negative is equal to negative 2 times the square root of 2 this one statement is equivalent to this right here because we have this plus or minus here this or statement let me solve all of these simultaneously so if I subtract 1 from both sides of this equation what do I have on the left-hand side I'm just left with 4 X on the right hand side I have you can't really mathematically I mean you can do them if you had a calculator but I'll just leave it as negative 1 plus or minus the square root or 2 times the square root of 2 that's what 4 X is equal to if we did it here is two separate equations same idea subtract 1 from both sides of this equation you get 4 X is equal to negative 1 plus 2 times the square root of 2 this equation subtract 1 from both sides for X is equal to negative 1 minus 2 times the square root of 2 this statement right here this statement right here is completely equivalent to these two statements now last step we just have to divide both sides by 4 so you divide both sides by 4 and you get X is equal to negative 1 plus or minus 2 times the square root of 2 over 4 now this statement is completely equivalent to dividing each of these by 4 and you get X is equal to negative 1 plus the square root of or 2 times the square root of 2 over 4 this is one solution and then the other solution is X is equal to negative 1 is to root of 2 all of that over for that statement and these two statements are equivalent and if you want I encourage you to let's substitute one of these back end just so you feel confident that something is Bizzaro as one of these expressions can be a solution to a nice vanilla looking equation like this so let's substitute it back in 4 times X well 4 times negative 1 plus 2 root 2 over 4 plus 1 squared minus 8 is equal to 0 now these fours cancel out so you're left with negative 1 plus 2 root 2 plus 1 squared minus 8 is equal to 0 this negative 1 and this positive 1 cancel out so you're left with 2 roots of 2 squared minus 8 is equal to 0 and then what you're going to have here when will you see you have two roots so when you square this you get 4 times 2 minus 8 is equal to is equal to 0 which is true 8 minus 8 is equal to 0 and if you try this one out you're going to get the exact same answer let's do another one like this remember these are special forms where we have squares of binomials in our expression and we're going to see that the entire quadratic formula is actually derived from a notion like this because you can actually turn any you can turn any quadratic equation into into a a perfect square equaling something else we'll see that two videos from now but let's get a little warmed up just seeing this type of thing so let's say you have x squared minus 10x plus 25 is equal to 9 now once again your temptation and it's not a bad temptation would be to subtract 9 from both sides so you get a 0 on the right hand side but before you do that just to inspect this really fast and say hey is this is this just maybe a perfect square of a binomial and you see well what two numbers when I multiply them by when I multiply them I get positive 25 and when I add them I get negative 10 and hopefully negative 5 jumps out at you so this is this expression right here is X minus 5 times X minus 5 so this left-hand side could be written as X minus 5 squared the right-hand side is still 9 and I want to really emphasize I don't want this to ruin all of the training you've gotten done factoring so far we can only do this when this is a perfect square if you got like X minus 3 times X plus 4 and that would be equal to 9 that would be a dead end you wouldn't be able to really do anything constructive with that only because this is a perfect square can we now say X minus 5 squared is equal to 9 and now we can take the square root of both sides so we could say that X minus 5 is equal to plus or minus 3 add 5 to both sides of this equation you get X is equal to 5 plus or minus 3 or X is equal to what's 5 plus 3 well X could be 8 or X could be equal to 5 minus 3 or X is equal to 2 now we could have done this equation this quadratic equation that's the traditional way the way you were tempted to do it what happens if you subtract 9 from both sides of this equation you'll get x squared minus 10x and what's 25 minus 9 25 minus 9 is 16 and that would be equal to 0 and here this would be just a traditional factoring problem with the type that we've seen in the last few videos what two numbers when you take their product you get positive 16 and when you take their when you sum them you get negative 10 and maybe negative 8 and negative 2 jump into your brain so we get X minus 8 times X minus 2 is equal to 0 and so X could be equal to 8 or X could be equal to 2 that's the fun thing about algebra you can do things in two completely different ways but as long as you do them in algebraically valid ways you're not going to get different answers let's and on some level if you recognize this this is easier because you didn't have to do that little game in your head in terms of oh what two numbers when you multiply them get 16 when you add them you get negative 10 here you just said okay this is X minus 5 I guess you did have to do it you have to say Oh 5 times 5 is 25 and negative 10 is negative 5 plus negative 5 so I take that back you still have to do that little that little game in your head so let's do let's do another one let's do one more of these just to really get ourselves nice and warmed up here so let's say we have let's let's do let's do one that's let's do this let's do it say we have x squared plus 18x plus 81 is equal to is equal to 1 so once again we can do it in two ways we could subtract one from both sides or we could recognize that this is X plus 9 times X plus 9 this right here 9 times 9 is 81 9 plus 9 is 18 so we can write our equation as X plus 9 squared is equal to 1 take the square root of both sides you get X plus 9 is equal to plus or minus the square root of 1 which is just 1 so X is equal to subtract 9 from both sides negative 9 plus or minus 1 and that means that X could be equal to negative 9 plus 1 is negative 8 or X could be equal to negative 9 minus 1 which is negative 10 and once again you could have done this the traditional way you could have subtracted 1 from both sides and you would have gotten x squared plus 18x plus 80 is equal to 0 and you to AG 8 times 10 is 80 8 plus 10 is 18 so you get X plus 8 times X plus 10 is equal to 0 and then you would get X could be equal to negative 8 or X could be equal to negative 10 that was good warm up now I think we're ready to tackle completing the square