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let's solve some quadratic equations by factoring so let's say I had x squared plus 4x is equal to 21 now your impulse might be to try to factor out an X and somehow set that equal to 21 and that will not lead you to good solutions you'll probably end up doing something that's not justified what you need to do here is put the entire quadratic expression on one side of the equation will do it onto the left-hand side so let's put let's subtract 21 from both sides of this equation the left-hand side then becomes x squared plus 4x minus 21 and then the right-hand side will be equal to 0 and the way you want to solve this this is a quadratic equation we have a quadratic expression being set equal to 0 the way you want to solve this is you want to factor them and say ok each of those factors could then be equal to 0 so how do we factor this well we saw in the last video that when you have to figure out two numbers whose product is equal to negative 21 and whose sum is equal to 4 so this would be a plus B would be have to be equal to 4 since their product is negative they have to be of different signs and so let's see the number that jumps out at me is 7 and 3 if I have negative 7 and positive 3 I would get negative 4 so let's do positive 7 and negative 3 so the a and B positive 7 and negative 3 when I take the product I get negative 21 when I take their sum I get positive 4 so I can rewrite this equation here I could rewrite it as X plus 7 times X minus 3 is equal to 0 and now I can solve this by saying look I have two quantities their product is equal to 0 that means that one or both of them have to be equal to 0 so that means that X plus 7 is equal to 0 that's an X or X minus 3 is equal to 0 I could subtract 7 from both sides of this equation and I would get X is equal to negative 7 and over here I can add three to both sides of this equation and I'll get X is equal to 3 so this both of these numbers are solutions to this equation you could try it out if you do 7 7 negative 7 squared is 49 49 negative 7 times 4 is minus 28 or negative 28 and that does indeed equal 21 and I'll let you try it out with the positive 3 actually let's just do it 3 squared is 9 plus 4 times 3 is 12 9 plus 12 is indeed 21 let's do a bunch more examples let's say I have x squared plus 49 is equal to 14x once again whenever you see anything like this get all of your terms on one on one side of the equation and get a 0 on the other side that's the best way to solve a quadratic equation so let's subtract 14x from both sides we could write this as x squared minus 14x plus 49 is equal to zero I'm going to see 14x minus 14x is 0 this quantity minus 14x is this quantity right there now we just have to think about what two numbers when I take the product I'm going to get 49 and when I take their sum I'm gonna get negative 14 so one they have to be the same sign because this is a positive number right here and they're both going to be negative because their sum is negative and there's something interesting here 49 is a perfect square it's factors are one seven and 49 so maybe seven will work or even better maybe negative seven will work and it does negative 7 times negative seven is times negative seven is 49 and negative seven plus negative 7 is negative 14 we have that pattern there where we have two times the number and then we have the number squared this is a perfect square this is equal to X minus seven times X minus seven it's equal to zero don't want to forget that or we could write this as X minus even squared is equal to zero so this was a perfect score this was a perfect square of a binomial and if X minus 7 squared is equal to 0 take the square root of both sides you'll get X minus seven is equal to zero I mean we could say X minus seven is zero or X minus seven is zero but that'd be redundant so we just get X minus seven is zero add 7 to both sides and you get X is equal to seven only one solution there let's do another one let's do another one in pink another one in pink let's say we have x squared minus 64 is equal to zero now this looks interesting right here this looks interesting you might have already you might already the Bell might be ringing in your head on how to solve this this has no X term but we could think of it as having an extra me could rewrite this as x squared plus 0x minus 64 so in this situation we could say okay what two numbers when I multiply them equal to 64 and when I add them they equal zero and then when I take their product I'm getting a negative number right this is a times B it's a negative number so that must mean that they have opposite signs so that must mean they have opposite signs when I add them I get zero that must mean that a plus minus B is equal to zero or that a is equal to B that we're dealing with the same number what we're essentially dealing with the same number there are the negatives of each other so what can it be well if we're doing the same number and their negatives of each other if they're if we're dealing with negatives of each other well the 64 is exactly 8 squared but it's negative 64 so maybe we're dealing with 1 negative 8 and we're doing with 1 positive 8 and if we add those two together we do indeed get to zero so this will be X plus or X minus 8 times X plus 8 now you don't always have to go through this process I did here you might already remember that if I have a plus B times a minus B then that's equal to a squared minus B squared so if you see something that fits the pattern a squared minus B squared you can immediately say oh that's going to be a plus B a plus B a is X B is eight times a minus B let's do a couple more of just general problems I don't tell you what type these are going to be let's say we have X let me switch colors it's getting monotonous let's say we have x squared minus 24x plus 144 is equal to zero well 144 it's conspicuously 12 squared and this is conspicuously 2 times negative 12 or this is considered to be negative 12 squared so this is negative 12 times negative 12 this is negative 12 plus negative 12 so this expression can be re-written as X minus 12 times X minus 12 or X minus 12 squared and we're going to set that equal to 0 this is going to be 0 when X minus 12 is equal to 0 you could say either of these could be equal to 0 but they're the same thing add 12 to both sides of that equation and you get X is equal to 12 and I just realized this problem up here I factored it but I didn't actually solve the equation so this has to be equal to 0 let's take a step back to this equation up here and the only way that this thing over here will be 0 is if either X minus 8 is equal to 0 or X plus 8 is equal to 0 so add 8 to both sides of this you get X could be equal to 8 subtract 8 from both sides of this you get X could also be equal to negative 8 so hopefully let's do one more just to really really get the point drilled in your head let's do one more let's say we have 4x squared minus 25 is equal to 0 so you might already see the pattern this is an a squared that's an a squared this is a B squared we have the pattern of a squared minus B squared where in this case a would be equal to 2x right this is 2x squared and B would be equal to 5 so if you have a squared minus B squared a squared minus B squared this is going to be equal to a plus B times a minus B in this situation that means that 4x squared minus 25 is going to be 2x plus 5 times 2x minus 5 and of course that will be equal to 0 and this will only be equal to 0 if either 2x plus 5 is equal to 0 or 2x minus 5 is equal to zero and then we can solve each of these subtract 5 from both sides you get 2x is equal to negative 5 divide both sides by 2 you could get one solution is negative 5 halves over here add 5 to both sides you get 2x is equal to positive 5 divide both sides by 2 you get X could also be equal to positive 5 halves so both of these satisfy that equation up there