In Exercises 1–38, solve each radical equation.
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√x - 4 ...

Question

In Exercises 1–38, solve each radical equation.
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√x - 4 + √x + 4 = 4

Accepted Answer

Welcome back. I am so glad you're here. We're told in the given radical equation to solve for X. Our given radical equation is the square root of X minus 16 plus the square root of X plus 16 equals eight. Our answer choices are answer choice. A, the solution set six, answer choice B, the solution set 16, answer choice C, the solution set 20 and answer choice D the solution set 36. All right. Typically when we're solving for X, we want to get X by itself on one side of the equal sign and all the numbers on the other side of the equal sign. But this time we have two XS and they are both trapped underneath radical signs. So we've got a little bit of a different strategy here. We need to isolate the radicals and then get rid of the radicals. Unfortunately, we have to do this one at a time. So we can start by isolating the first radical, the square root of X minus 16. So we need to move the second radical to the right hand side of the equal sign because it's being added, we want to do the opposite of that, we will subtract the square of X plus 16. And what we do to one side, we do to the other. So on the left, the positive square of X plus 16 minus the square rot of X plus 16 is zero. So we just have the square rut of X minus 16 on the left and on the right, we have eight minus the square rut of X plus 16. So now that we've isolated one of the radicals the way to rescue that ax to undo that radical when we're squaring something we want, or we're taking the square root of something we want to square it. And what we do to one side we need to do to the other as well. That's where this one gets fun. So on the left, the square rut of X minus 16 squared is just X minus 16. That side's lovely now and on the right, we need to take eight minus the square rut of X plus 16 and square that in order to do that, we need to foil it out. So let's come over to the side and we'll take the quantity of eight minus the square root of X plus 16 multiplied by the quantity of eight minus the square root of X plus 16. Yep, this one's so fun. Now we'll foil this out. So we take our firsts, eight multiplied by eight is a positive 64. Now our outsides eight multiplied by the negative square root of X plus 16 is going to be a negative eight square rut X plus 16. Our insights, we have the negative square rut of X plus multiplied by eight, which will again be negative eight square rut of X plus 16. And then our lasts, we have the negative square root of X plus 16 multiplied by the negative square ret of X plus 16, which will give us a positive square rot of X plus 16 squared. So that 64 that just hangs out for a little bit. And now we can combine our inside like terms this negative eight square it of X plus 16 minus another eight square it of X plus sixteens, which will be negative 16 square rats of X plus 16. And now taking the square root of X plus 16 squared, that will give us just a plus X plus 16. And we can combine the like terms here. We've got this 64 plus 16 which will be a positive 80. And then we can have, well, we could put this in a totally different order, but we can keep it for now as 80 plus X and then minus 16 square rut of X plus 16. And that'll do for now. So the right hand side of this equation is 80 plus X minus 16 square root X plus 16. So now back to what we started doing, we still have to rescue this other X from this other radical. So we need to get that radical isolated again. Let's start off by moving this X and this over to the left hand side of the equation. So because the X is being added, we subtract X from both sides and because the 80 is positive will subtract 80 from both sides. And on the left where we have X minus X, that's zero. Lovely. And where we have negative 16 minus 80 that's a negative 96. And that's equal to on the right, the 80 minus 80 is zero and the X minus X is zero. So just the negative 16 square root X plus 16. And now we need to still get that radical isolated. We will divide both sides by negative 16. And so on the left, we have a negative 96 divided by a negative 16. That's going to be a positive six that's equal to on the right, negative 16 divided by negative 16 is just one. So we have finally isolated this radical. It is the square root of X plus 16 by itself over here. And again, what we do to get that get rid of that radical is we will square and what we do to one side, we do to the other as well. So we're squaring both sides and on the left six squared is a positive that's equal to on the right, the square root of X plus 16 squared is just X plus 16. We almost have X by itself. Yay. Right now, we just subtract 16 from both sides of the equation. In order to get X by itself on the left, 36 minus 16 is a positive 20. And on the right, the 16 minus 16 is zero. So we just have X so we have our solution X equals 20. We look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

In Exercises 1–38, solve each radical equation. √x - 4 + √x + 4 = 4

Key Concepts

Radical Equations

Isolating the Radical

Extraneous Solutions

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