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in this video we're going to talk about extraneous solutions if you've never heard the term before I encourage you to review some videos on Khan Academy on extraneous solutions but this is a bit of a refresher it's the idea that you do a bunch of legitimate algebraic operations you get a solution or some solutions at the end but then when you test it in the original equation it doesn't satisfy the original equation and so the key of this video is why do extraneous solutions even occur and it all is due to the notion of reversibility there's certain operations in algebra that you can do in one direction but you can't and it'll always be true in one direction but it isn't always the true it is always true in the other direction and I'll show you those two operations one is squaring and the other is multiplying both sides by a variable expression so let's see the example of squaring and then we're going to see it in an actual scenario where you're dealing with an extra with an extraneous solution so we know for example that if a is equal to B I could square both sides and then a squared is going to be equal to B squared but the other way is not true for example if a squared is equal to B squared it is not always the case that a is equal to B what's an example that shows that this is not always the case actually pause the video try to think about it well negative 2 squared is indeed equal to 2 squared but negative 2 is not equal to 2 so this shows that you can square the both sides that both you can square both sides of an equation and get and deduce something that is true but the other way around is not necessarily going to be true another non reversible operation sometimes is multiplying both sides by a variable expression so multiplying both sides actually let me guess confused looks like an X multiply both sides by variable I'll just write variable but but it could be a variable expression as well for example we know that if a is equal to B that if we multiply both sides by a variable that's still going to be true X a is going to be equal to X B but the other the reverse isn't always the case if X a is equal to X B is it always the case that a is equal to B well the simple answer is no and I always encourage you pause this video and see if you can find an example where this doesn't work well what if the very if a was 2 and B is 3 and the variable X just happened to take on the value 0 so we know that 0 times 2 is indeed equal to 0 times 3 but 2 is not equal to 3 now how does all of this connect to the extraneous solutions you've seen when you're solving radical equations or when you're solving some rational or equations with rational expressions on both sides well let's look at an example let's look let's solve a radical equation if I wanted to solve the equation the square root of 5x minus 4 is equal to X minus 2 a typical first step is hey let's get rid of this radical by squaring both sides so I am going to square both sides and then I'm going to get 5x minus 4 is equal to x squared minus 4x plus 4 once again if this looks completely unfamiliar to you we go into much more depth in other videos where we introduce the idea of radical equations and let's see we can subtract 5x from both sides we can add 4 to both sides I'm just trying to get a 0 on the left hand side and so I'm going to be left with 0 is equal to x squared minus 9x plus 8 or 0 is equal to X minus 8 times X minus 1 or we could say that X minus 1 is equal to 0 or X minus 8 is equal to 0 we get x equals 1 or x equals 8 so let's test these solutions if x equals 8 we would get in color-coded a little bit 4x equals eight if I test it in the original equation I get the square root of 36 is equal to 6 which is absolutely true so that one works but what about x equals one I get the square root of 5 times 1 minus 4 is 1 is equal to 1 minus 2 which is equal to negative 1 that did not work this right over here is an extraneous solution if someone said what are all the X values that satisfy this equation you would not say x equals 1 even though you got there with legitimate algebraic steps and the reason that is true is actually pause this video look back for which of these steps this x equals 1 X equal 1 still work and what step does it not work well you'll see that x equals 1 works for all of these equations below this purple line it just doesn't work for the square root of 5 minus 4x is equal to X minus 2 in fact you can start with X minus 1 and then you could deduce all the way up to this line here but the issue here is that squaring is not a reversible operation this is analogous to saying hey we know that a squared is equal to B squared we know that this is equal to this but then that doesn't mean that a is necessarily equal to B for x equals 1 and we could do the same thing with a rational or an equation that deals with rational expressions so for example we might have to deal with it let me make sure I have some space here if I had to solve x squared over X minus 1 is equal to 1 over X minus 1 the first thing I might want to do is multiply both sides by X minus 1 so multiply multiplied by X minus 1 I notice I'm multiplying both sides by a variable expression so we have to be a little bit conscientious now but if I multiply both sides by X minus 1 I'm going to get x squared is equal to 1 or I could say that x equals 1 or X is equal to negative 1 but we could test these for x equals 1 if I go up here I'm dividing by 0 on both sides so this is an extraneous solution the key here is that we multiplied both sides by a variable expression in this case we multiplied both sides by X minus one you can do that you can multiply both sides by a variable expression and it is a legitimate algebraic operation it's completely analogous to what we saw right over here just because zero times two is equal to zero times three does not mean that two is equal to three it's completely analogous because we multiplied by a variable expression that actually takes on the value zero when X is equal to one so the big takeaway here is hopefully you understand why extraneous solutions happen a little bit more when you square when you multiply both sides by variable expression completely legitimate as long as you do it properly but it's not always the case that the reverse is true you could add or subtract anything from both sides of an equation and that's always going to be reversible and so you don't that's not going to lead to extraneous solutions you can multiply or divide by a nonzero constant value that's also not going to lead to anything Shady but if you're squaring both sides or multiplying both sides by a variable expression you should be a little bit careful