 In this video, we provide the solution to question number 18 from the practice final exam from math 1050 We have a radical equation the square root of x minus the square root of x minus 5 is equal to 1 We need to find all real solutions so we can ignore any complex solutions if there's any just real solutions here The strategy we want to employ here is we want to separate the radicals So just give one square root on let's say the left-hand side all by itself because of the sign here I'm actually just gonna add the square of x minus 5 to both sides So we get the square root of x is equal to 1 plus the square root of x minus 5 like so We then are going to proceed to square both sides because squaring will get rid of the square root on the left-hand side Now one has to be careful here when you square both sides of the equation. You're now introducing Party crashers to this equation. That is when we're done We might have solutions that actually aren't solutions. They were imposters So we need to make sure we check our answers at the end generally speaking We should always check our answers to make sure we have a correct solution But with equations involving square roots It's imperative because even if you did all the algebra correctly if you don't check your solutions Then you can't tell the difference between a real solution or one of these party crashers On the left-hand side if you square the square root of x you'll end up with just an x on the right-hand side You do have to foil this thing because this is 1 plus the square root of x minus 5 times 1 plus the square root of x minus 5 So when you foil that out you'll get 1 plus 2 times the square root of x minus 5 And then you add to that you'll get the square root of x minus 5 square, which is x minus 5 itself So my recommendation here is going to be combined like terms. Let's subtract x from both sides And then we have this negative 5 plus 1 like so in that situation The x's are actually going to cancel each other out x minus x x minus x We have then 0 is equal to negative 4 plus 2 times the square root of x minus 5 like so Let's move the negative 4 to the other side So we end up with 4 is equal to 2 times the square root of x minus 5 Since both sides are visible at 2 it'd be nice to get rid of that coefficient so divide both sides by 2 We then have the square root of x minus 5 is equal to 2 Much like we did before we have to square both sides now since we've already squared both sides We're already cautious of these party crashers We don't have to worry about it again. I mean because the problem is already there now On the left-hand side the square root of x minus 5 will just be when you square let's get x minus 5 on the right-hand side You get 4 Adding 5 to both sides. We're looking at x equals 9. That's our potential solution But again, we have to be cautious We have to check it if we don't check it we have some problems here So let's put into the original equation if you put in a 9 and a 9 what happens on the left-hand side The left-hand side is going to give you the square root of 9 Minus the square root of 9 minus 5 like so the square root of 9 is 3 9 minus 5 is 4 whose square root is 2 that is in fact equal to 1 So we do get the we do get that 9 is an authentic solution and that was the only number we found So therefore we report that the solution to this equation was x equals 9