 Let's take a look at a few more root equations. So our ability to solve root equations is based on the idea that if the nth root of x is equal to a number, then a to the power n is equal to x. So if I have square root of x plus 5 equals x minus 1, I can start by squaring both sides. And let's clean this up a little bit algebraically, expanding. And because we get a quadratic equation, let's get all of our terms on one side. So after all the dust settles, our original equation has been transformed into a quadratic equation. Now we can solve this quadratic equation any way we like. And today we like the quadratic formula. So let's use that to find our solutions, which are going to be 4 or negative 1. Remember, we always need to check to see if these are actually solutions. So we'll check x equals 4. We'll substitute that into our original equation. And since the square root of 9 is actually equal to 3, this is true. So x equals 4 is a solution. We'll also check x equals negative 1. It's important to remember that principal square root of A always indicates a non-negative number. And so this statement, principal square root of 4 equals negative 2 is false. So x equals negative 1 is not a solution. And we should indicate this when we write up our answer. All right, let's take another example. Now remember, we know what to do in the case of root equals, so we need to isolate the root. So right now we have square root plus 1. Let's get rid of that plus 1. And if we subtract 1 from both sides, we now get square root equals. We can square both sides and simplify. And because this is a quadratic equation, let's get everything on one side. And that produces a corresponding equation. Solve the equation any way you like, which gives us solutions x equals 5 and x equals 1. And now let's check our two solutions. Checking x equals 5 by substituting it into the original equation. And we get a true statement, so x equals 5 is a solution. We'll also want to check x equals 1 by substituting it into the original equation. And this is also true, so x equals 1 is also a solution. So both our solutions are solutions to the original equation. Or we could take a look at another example. This is already in the form square root equals, so we'll square both sides. And we'll solve this somehow. And remember, if you don't write it down, it doesn't exist. So let's summarize what we've done so far. And of course we'll circle our answer. And because we did this, our answer is completely wrong. Because any time we're dealing with a root equation, we always have to check to make sure our solutions satisfy the original equation. So let's rewind. Checking our solution, x equals 2, does not make the original equation a true statement. So x equals 2 is extraneous. We'll check x equals 1. And again, x equals 1 also makes a false statement. So x equals 1 is also extraneous. But these were the only two solutions that we had. And since both are extraneous, this equation has no solution. How about something like this horrible thing? And a useful idea to keep in mind is that anything you can do once, you can do multiple times. So in other problems, we isolated the square root and got rid of it by squaring both sides. Well, let's try it. So the left-hand side has two square roots. So let's get rid of one of them. So maybe we'll subtract this square root 4x minus 3, and we do have to do the same thing to both sides. So now I have square root equals mess. Well, I can square both sides, expanding the right-hand side, and we can do a little bit of algebraic cleanup. And remember, if it's not written down, it didn't happen. But wait, we still have a square root in this equation. Well, let's isolate the remaining root and square. So we'll rewrite the equation so the square root term is on the left, square both sides, and do a little bit more cleanup. And again, if it's not written down, it didn't happen. Now we have a quadratic equation, so solve this by using your favorite method. Now maybe your favorite method is having the solutions fall out of the sky and hit you on the head, but a slightly less hazardous approach can use things like completing the square or the quadratic formula. And again, the important thing to do with any sort of root equation is to always check to make sure your solution satisfied the original equation. And when we do that, we find that x equals 1 is a solution, but x equals 13 is an extraneous solution.