8.7 Radical equations

Before I get into the radical equations, there is something very important I have to get out of the way. Square these two out:

How’d it go? If you got six for the first answer and five for the second, stop! Go back and look again, because those answers are not right. (If you don’t believe me, try it on your calculator.) When you’ve got those correctly simplified (feel free to ask—or, again, check on your calculator) then go on.

Now, radical equations. Let’s start off with an easy radical equation.

This next one is definitely trickier, but it is still in the category that I call“easy”because there is still no x under the square root.

Now, what if there is an $x$ under the square root? Let’s try a basic one like that.

What did you get? If you said the answer is three: shame, shame. The square root of 3 isn’t 9, is it? Try again.

OK, that’s better. You probably guessed your way to the answer. But if you had to be systematic about it, you could say“I got to the answer by squaring both sides.”The rule is: whenever there is an $x$ under a radical, you will have to square both sides. If there is no $x$ under the radical, don’t square both sides.

It worked out this time, but squaring both sides is fraught with peril. Here are a few examples.

Hey, what happened? When you square both sides, you get $x=81$ , just like before. But this time, it’s the wrong answer: $\sqrt{81}$ is not $-9$ . The moral of the story is that when you square both sides, you can introduce false answers. So whenever you square both sides, you have to check your answers to see if they work. (We will see that rule come up again in some much less obvious places, so it’s a good idea to get it under your belt now: whenever you square both sides, you can introduce false answers! )

But that isn’t the only danger of squaring both sides. Check this out…

Hey, what happened there? When you square the left side, you got (I hope) $x+4\sqrt{x}+4$ . Life isn’t any simpler, is it? So the lesson there is, you have to get the square root by itself before you can square both sides . Let’s come back to that problem.

Whew! Much better! Some of you may have never fallen into the trap—you may have just subtracted the two to begin with. But you will find you need the same technique for harder problems, such as this one:

Solve #10 and check your answers…