 Another thing we can do is we can talk about equations that have radicals in them. And this is not somebody like Abby Hoffman, but rather it's an equation that involves a root, generally a square root, but we can look at other roots as well. So in general, if I have the square root of a equals b, then I know that a is equal to b squared. And conversely, if I have a equals b squared, I can reverse that and get b is equal to the square root of a. And what this means is that if I have an equation that involves a radical, some sort of a root, I can use algebra to isolate that radical and then square both sides. If it's a quadratic, I can square both sides and get rid of the square root. Now, one thing we have to be careful with this symbol always means the positive number whose square is a. And what that means is that I have to make sure that I always ended up with a positive number here. And when I square both sides, I lose sine information. So it's possible that when I'm solving a radical equation, I may end up with what are called extraneous solutions, solutions to this equation that won't actually work as solutions to the original equation. And because of that, it's important to always test your solutions to verify that they really are solutions. For example, let's say I want to solve square root of x minus five equals x minus seven. So the first thing I want to do is I want to get my equation in the form square root equals. Well, I have it square root equals stuff. So this is already in the form that I want it to be in. So now I can square both sides, square both sides, and that will get rid of the square root over on the left hand side. Over on the right hand side, I have x minus seven squared. I'll expand that out. And let's see, I have a quadratic equation. So I'll want to get all of my terms over to one side. And let's see, I'll subtract x add five, subtract x add five to both. Over on the left hand side, everything is gone over on the right hand side. That's x squared minus 15x plus 54. And here I have the quadratic equation. So I'll find the solutions using the quadratic formula. And I'll substitute in my values. And after all the dust settles, I have my solutions nine or six. Now it's important to understand at this point that these are potential solutions, x could be nine or x could be six. However, because I squared both sides of my radical equation, it's possible that one or both of these solutions won't actually solve my original equation. So I need to check. So we'll check x equals nine. If x equals nine, the left hand side square root nine minus five, the right hand side nine minus seven. And I want to determine whether these two are equal. So I'll evaluate square root of four is two. And I have to decide is two equal to two. Well, that's a true statement. So x equals nine is a solution. Now I need to check the other potential solution x equals six. So I'll substitute that into my original equation. And I'll let the dust settle square root of one is equal to one. And this is a false statement. One is not equal to negative one. So x equals six is an extraneous solution. And we should make sure that we indicate that. So I have my potential solutions. But then I know that x equals six is extraneous and not a solution. Well, let's take a look at another example, square root of x plus three plus five equals two. And again, this is not in the form square of a equals b. So I have to transform it into that form. So I'll subtract five from both sides. I'll get square root of x plus three is equal to negative three. Now a little analysis goes a long way. At this point, if you remember the definition of square root, it is the positive number whose square is whatever the thing is inside. From that definition, square root can never be a negative number. Square root of anything cannot be a negative number. So no value of x will make this equation true. If you recognize that, and again that is from a little bit of analysis and remembering the definition of square root, from that information, this equation has no solution, which means that we can actually stop solving the problem right here and say there is no solution. If we don't recognize that, if we don't do the analysis, if we don't remember what the definition of square root is, we can do a little bit more work. We can square both sides. We can solve for x, x equals six. And I find x equals six is a potential solution. We have to check our solutions in the original equation. So x equals six is a potential solution. So I'll substitute that into my original equation. And I end up with a false statement. 3 plus 5 is definitely not equal to 2. And so that says that x equals 6 is not a solution. It's an extraneous solution. And since it's the only potential solution, this equation has no solutions. And again that's something we actually could have realized back here. If we remembered the definition of square root, if we did a little bit of analysis, we would have saved ourselves all of this extra work by doing those things. So a little bit of analysis goes a long way. The definitions are really important.