- In this video, we're walking to talk aboutextraneous solutions. If you've never ever heard the ax before, i encourage you to testimonial some videos on khan Academy ~ above extraneous solutions. But just together a little of a refresher,it's the idea the you perform a bunch of legitimatealgebraic operations, you gain a systems orsome solutions at the end, however then when you test itin the original equation, the doesn't fulfill the original equation. And also so the an essential of this video is why do extraneous solutions also occur? and also it every is due to thenotion of reversibility. There's details operations in algebra the you deserve to do in onedirection, yet you can't, and it'll always be true in one direction, however it isn't always truein the other direction. And I'll display you those two operations. One is squaring and also the otheris multiply both sides by a variable expression. For this reason let's simply see the instance of squaring, and also then we're going to seeit in an actual scenario whereby you're handle withan extraneous solution. So we know, for example,that if a is equal to b, I could square both sides and also then a squared is goingto be same to b squared. However the other means is no true. For example, if a squaredis equal to b squared, that is not constantly the casethat a is equal to b. What's a example that mirrors that this is not constantly the case? Actually, stop the video,try to think about it. Well, negative two squared is certainly equal to two squared, but an adverse two is not equal come two. So this reflects that girlfriend cansquare both sides of one equation and deduce something the is true, however the other means around is no necessarily walk to be true. An additional non-reversible operation periodically is multiply both sidesby a change expression. So multiplying both sides, actually, favor I said,that will certainly make united state confused, looks choose an x. Main point both sides by variable. I'll simply write variable, however it could be a variableexpression together well. Because that example, us knowthat if a is same to b, that if us multiply bothsides by a variable, that's still going to be true. Xa is going to be equal to xb. But the other, the reverse,isn't always the case. If xa is same to xb, is it constantly the casethat a is equal to b? Well, the basic answer is no, and I constantly encourageyou, pause this video and view if you can find an example where this doesn't work. Well, if a was two and b is three and also the change x simply happenedto take it on the worth zero. For this reason we understand that zero times two is indeed equal to zero times three, however two is no equal to three. Now, how does all this connect to the extraneous solutions you've seen once you were fixing radical equations or as soon as you were addressing some rational, or equations v rationalexpressions ~ above both sides? Well, let's look at an example. Let's solve a radical equation. If I want to fix the equation the square source of 5 x minus four is same to x minus two, a typical first step is, hey,let's eliminate this radical by squaring both sides. So I'm going come square both sides, and then I'm going toget 5 x minus 4 is same to x squaredminus four x add to four. As soon as again, if this lookscompletely unfamiliar come you, we get in much more depth in other videos where we present theidea that radical equations. And let's see, we deserve to subtractfive x indigenous both sides. Us can include four come both sides. I'm simply trying to obtain azero top top the left hand side. And so I'm going to be left with zero is same to x squaredminus nine x plus eight, or zero is equal to x minuseight times x minus one, or we can say that xminus one is equal to zero, or x minus eight is same to zero. We gain x equals one or x equates to eight. Therefore let's check these solutions. If x equates to eight, we would get, and I'll color code the a tiny bit, because that x equates to eight, if i testit in the initial equation, I obtain the square rootof 36 is equal to six, i beg your pardon absolutely true, so that one works. However what around x amounts to one? I obtain the square source of fivetimes one minus 4 is one is same to one minus two,which is equal to negative one. That did no work. This best over below isan extraneous solution. If who said, what room all the x worths that satisfy this equation,you would not speak x equates to one, even though you acquired there withlegitimate algebraic steps. And also the reason that is true is, actually, pause this video, look back. Because that which of this stepsdoes x equal one tho work and what step does it no work? Well, you'll watch that x amounts to one functions for all of these equationsbelow this purple line. It simply doesn't occupational for the square source of five minus 4 x is equal to x minus two. In fact, you might start v x minus one and also then you can deduce allthe method up to this heat here. But the worry here is that squaring is not a reversible operation. This is analogous to saying, hey, we know that a squaredis equal to b squared. We understand that this is equal to this. However then that doesn't median thata is necessarily equal to b because that x equals one. And also we could do the samething with a rational, or an equation the dealswith reasonable expressions. So, for example, wemight have to deal with, and also let me make sure Ihave some room here. If I had to solve xsquared over x minus one is same to one over x minus one, the very first thing I can wanna perform is main point both sides by x minus one. So main point by x minus one. Now, notice I'm multiply both sides by a change expression, therefore we need to be a small bit contentious now, however if I'd main point bothsides through x minus one, I'm going to acquire xsquared is same to one, or I can say the x amounts to one or x is equal to negative one. Well, we might test these. For x amounts to one, if i go increase here, I'm separating by zero top top both sides. For this reason this is one extraneous solution. They crucial here is thatwe multiply both sides by a variable expression. In this case, us multipliedboth political parties by x minus one. You can do that. You have the right to multiply both sidesby a variable expression, and it is a legitimatealgebraic operation. It's fully analogous towhat we saw appropriate over here. Just because zero times twois same to zero times three does not median that 2 is same to three. It's completely analogous since we multipliedby a variable expression that in reality takes on the worth zero when x is same to one. Therefore the large takeaway hereis, hopefully you understand why extraneous solutionshappen a tiny bit more. As soon as you square, whenyou main point both political parties by a change expression,completely legitimate as lengthy as you do it properly, however it's not constantly the situation that the reverse is true. Friend could include or subtract anything native both sides of an equation, and that's constantly going to be reversible. And also so that's no going tolead come extraneous solutions.

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You can multiply or division by a non-zero continuous value. That's likewise not walking tolead to anything shady, yet if you're squaring both political parties or multiply both sidesby a variable expression, you have to be a small bit careful.