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Video transcript

we're on problem 48 it says if x squared is added to X the sum is 42 so let's just write that out if x squared is added to X the sum is equal to 42 which of the following could be the value of x so essentially just want us to solve this equation so the easiest way to do it is to write it as a as a quadratic equaling 0 and then factoring it so we could write this as x squared plus X minus 42 is equal to 0 and let's think what two numbers when I add them equal 1 and when I multiply them equal minus 42 and the fact that when I multiply them equals minus 42 tells me that one of them has to be positive and one of them has to be negative that's the only way that you're gonna when you multiply two numbers you're gonna have a negative number so one of them has to be positive one of them has to be negative and so when we're adding a positive and a negative you're really finding the difference between the two so the difference between the two numbers has to be one and the product has to be 42 and I know this when I see 42 I immediately think oh 6 + 7 6 times 7 is 42 and since when you add them you get a positive 1 7 is probably the positive one and and the -6 or 6 is probably the negative one so let's try it out X plus 7 times X minus 6 equal to 0 and right 7 times minus 6 is minus 42 and then 7 it's really 7 X plus minus 6 X is equal to positive x we could sing 7 plus minus 6 is equal to the coefficient on X which is 1 anyway that works and you could multiply this out and try it out and all everything I'm saying I'm not you know it's not some voodoo the reason why I I say that they have to add up to the 1 is because when you multiply this out that's what builds up this term the 7 times X plus the minus 6 times the other X that's what builds this term when you multiply it out right this term comes from the x times the X the minus 42 comes from the 7 times a minus 6 anyway now at this point we say okay well how do we get this we have two things when you multiply them equal to 0 well that means that one of them or both of them have to be equal to 0 so that means that X plus seven is equal to zero which would mean subtract seven from both sides which means X is equal to minus 7 or X minus six is equal to zero add 6 to both sides X is equal to 6 X would be 6 or minus 7 and they have one of the choices there which was choice a next problem 49 what quantity should be added to both sides of this equation to complete the square so when you complete the square you want the thing to just look like a up you want that what's ever on the left hand side to look like a perfect square and what do I mean by a perfect square so you know if I had X plus a squared that's equal to X plus a times X plus a and that's equal to x times X x squared x times that a so that's plus a X and now this a times this X so that's another ax plus this say times that a so plus a squared and that's equal to x squared plus we have two of these now plus 2ax plus a squared so essentially we want this we want the left-hand side to have this form so we say oh this is a perfect square we can say that's the same thing as X plus a squared so let's think about how we can do it if we have x squared minus 8x is equal to 5 and I put a space here for a reason because we want to add or subtract something here so it looks like a perfect square so think about it whatever it would when we have this format right in order for this thing to be a perfect square whatever this coefficient is right here this term right here has to be 1/2 of this squared right a squared is 1/2 of 2a squared so if we took half of minus 8 that's minus 4 right in this case if we said 2a is equal to 8 a would be minus 4 and so minus 4 squared is what it's plus 16 plus 16 and this is an equation so one thing you do to one side of an equation you have to do to the other side of the equation so you have to say that that is also equal to so you have to add 16 to both sides otherwise you're changing the equation anyway now hopefully recognize this is already a perfect square I mean you could look at this pattern up here or you could say okay if I add minus four to itself twice I get minus eight if I multiply it by itself it gets 16 so this is X minus four squared and that's equal to 25 and actually just so if you're curious and we did this in the Khan Academy we did a couple of videos on this this is how you prove the quadratic equation you essentially complete the square with you know arbitrary numbers a B and C and you get the quadratic equation and it's not you know we show it in ten minutes so it's not this you know impossibly hard thing to understand but if you have to they just want to know what to you what do you add to both sides of this equation right what quantity should be added to both sides of cick waiting to complete the square so the answer to this one was 16 but they just as equally could have said you know solve it by completing the square and you say oh X minus 4 squared is equal to 25 so X minus 4 is equal to plus or minus 5 and then you could say X is equal to plus or minus 5 plus 4 and then you could today okay which is 4 plus positive 5 is 9 4 plus -5 is or minus 1 anyway they didn't ask us that so we don't have to spend too much time thinking about it see or on problem 50 let me see problem 50 copy and paste 50 and 51 50 and 51 all right what are the solutions for that a quadratic equation x squared plus 6x is equal to 16 x squared plus 6x is equal to 16 and the temptation here is really the kind of try to solve it the way you do a linear equation I don't know factor out an X and I don't know to whatever else but the important thing to recognize is this is a quadratic equation and the easiest way to solve it is to put all the terms on one side and then get a zero on the other side and then either factor it or use the actual quadratic equation or you know complete the square or whatever you need to do so let's subtract 16 from both sides and you get x squared plus 6x minus 16 is equal to 0 right I just subtracted 16 both sides to get here and before just jumping into the quadratic equation let's see if we can factor it by inspection so what two numbers when I when I add them equal six and went positive six and when I subtract when I multiply them equal nine to sixteen and once again since it's the minus sixteen if you multiply two numbers you get a negative number they have to be different signs one has to be positive and one has to be negative and and and their difference will be six because once positive ones negative so let me think about it so if I had - what eight and two is equal to sixteen and there are six apart so if I did plus 8 and minus two right plus eight minus two is positive 6 so it's X plus 8 times X minus 2 and that really just takes a lot of practice you say ok what two numbers 16 ok 8 and 2 well they're gonna have to be different signs so but I have a positive 1 here so whichever number is larger is probably gonna be the positive one so positive 8 minus 2 yeah when you add them up they equal minus 6 yeah it works so you set that equal to zero and you say okay this has to be equal to zero or that has to be equal to zero so X is U so X is either equal to minus eight right if you say X plus eight is equal to zero then subtract 8 from both sides you get X is equal to minus eight I shouldn't have skipped that step but I'll do the step here or you could say X minus two is equal to zero add two to both sides you get X is equal to two you know what what X eat makes this term equal zero and you can look at it from inspection so X could either be minus eight or two and that is choice C problem 51 Leanne correctly solved the equation x squared plus 4x equals six by completing the square which equation is part of our solution okay so same thing we just have 2x squared plus 4x when you complete the square you gonna add something here so I'm gonna leave a little blank is equal to 6 so what could I add here that makes this expression look like a perfect square well we go to the pattern that we did a couple of problems ago whatever's here should be the square of half of this so for what half of that is 2 two squared is four so I should add 4 to that side bad for that's I'd have to add 4 to this side as well and now this 2 plus 2 is equal to 4 2 times 2 is equal to 4 so this is X plus 2 squared and I really want you get the intuition don't memorize the step for completing the square I want you to really understand why what you know this is the square of 1/2 of that we showed it in the beginning you know square a lot of a binomials and see for yourself but that's always going to be the case anyway so this is X plus 2 squared and it's going to be equal to 6 plus 4 is equal to 10 and that is choice B choice B I think we have time for one more one more problem problem 52 copied it and now I have pasted it Carter is solving this equation by factoring which expression could be one of his correct factors all right once again I like to I like to personally separate out the the number that goes into all of them and all these are just by 5 that just simplifies it in my head so if I divide all of these by 5 actually I could just divide both sides of this equation by 5 right 0 divided by 5 is 0 and then the left side divided by 5 becomes 2x squared minus 5x minus 5 X plus 3 is equal to 0 ok so this is 2x squared here so it's gonna be two numbers when you multiply them equal 3 and when you so let's think about this a little bit let me write it actually let me write it down here cuz I think I'll need more space 2x squared minus 5x plus 3 is equal to 0 and I just divide both side of this equation by 5 to get to this so let's see if we can what we can do here so we have a 2x squared here and they already kind of hinted us that we're gonna have an integer solutions that we can factor this so the intuition is that this is going to be 2x times you know plus something plus a times well what x times probably X right 2x times access to x-squared and that wouldn't be completely obvious if they didn't already tell us that you know we could factor this you might have to use a quadratic equation or something actually the quadratic equation wouldn't be something crazy to use here because you can just kind of plug in charge but let's see if we can get the intuition so it's going to be 2x plus something times X plus something else and if we were to multiply this out you get 2x times X is 2x squared as it should 2x times B is plus 2b X a times X is plus ax a times V is plus a B and so let's see what we get we get we get a so plus 2b plus a X plus a B 2x squared ok now we can do pattern matching right this was our original thing so two times B plus a has to be equal to write this term is the same thing as this term right here and that term has to be the same thing as that term right there so what two numbers so first of all I have a positive 3 here so I'm multiplying two numbers to get a positive 3 so they either have to be both positive or both negative right and then the other interesting thing is we have when I take two times one of them plus the other one I get a mind a negative number so the only way when you're dealing with negative numbers and you know when you just multiply times a positive and add them to each other you get another negative number is if they're if they're both negative right this told us they both have to be negative because this is positive and then since when you add them and I'm kind of you know without any negative signs you get a negative number you tells you that that has to be negative as well so let's see let's just try 3 and negative 3 negative 1 so if we have negative if negative 3 negative 1 yeah right yeah if B if B is B is equal to minus 1 and a is equal to minus 3 then 2 times minus 1 is minus 2 minus 3 right so B is equal to minus 1 and a is equal to minus 3 and so this is a bit of a of an art form here I mean there isn't like a plug and chug very mechanical way of doing this the quadratic equation is one but is the best way that at least I know how to do these without so we know what a and B is so it's 2x a is minus 3 2 X minus 3 times X plus B B is minus 1 so that's the factorization and what do they so 2 X minus 3 times X minus 1 which one they have this one right here 2 X minus 3 and I'm all out of time see in the next video