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In $3-38,$ solve each equation for the variable, check, and write the solution set.$$x+\sqrt{x+1}=5$$

$\{3\}$

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Chapter 3

REAL NUMBERS AND RADICALS

Section 8

Solving Radical Equations

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

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In $3-38,$ solve each equa…

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Okay, so here we have X plus. The square root of X plus one is equal to five so that isolate the square root. Let's go on. Let's subtract X from both sides. The equation so that we get the square root off X plus one is equal to all five minus acts. So now we go ahead and square both sides to kill off right to eliminate the radical or the square root side. So that's right. The square root of X plus one squared is just gonna be equal to X plus one. But then if we square the right side with five minus X squared means while five minus X, right times five minus acts that gives us a 25 and then a minus. Phi backs minus buybacks that's minus 10 X. And then you get a plus X squared. So we write this as well. X squared is equal five minus X squared is equal to X squared minus 10 X plus 25. Okay, so now we can go ahead and basically get a quadratic by getting by. Setting are quadratic equal to zero by subtracting right our ex term and R one and R one right to the other side of the equation to give us what we haven't x squared and then was attracting the actual 10 x minus X that's minus 11 X and then 25 minus one is while 24 So plus 24 okay is now equal to zero. So right, So we What we did here is we subtracted the X over to isolate the radical, and then we square both sides but squaring both sides Well, that eliminated the square root to just give us X plus one. But then squaring five minus X men, it meant we had to take by minus x times itself to give us right X squared minus 10 X plus 25 then studying our equation equals zero gives us X squared minus 11 X plus 24. And now can we go ahead and try and factor this? Well, um, well, what's gonna multiply? 24 um, eight times three right is 24. And when a plus three is 11 so I think was when x square, it has to be a next time the next. Right now 24 is gonna be an eight times a three. I have to somehow, any the same sign, right? Multiple to a positive 24. So everything is negative. That gives me an X squared minus three X minus eight X is minus 11 x and then plus 24. So this is FACTORED, and this is still gonna be, of course, equal to zero. So this means that if X minus eight times X minus three is equal to zero, either X minus eight is equal to zero or X minus three is equal to zero, so X minus eight is equal to zero. That means that X is equal to eight or if x minus music with zero, that would imply that X is equal to three. OK, so now let's go ahead. And that's check both of these answers in our original saloon and I was not present here, So X is equal to eight. What would we get? We get eight, Um, plus the square root of a plus one. So a plus one is what a plus one is You to nine. The square root of nine is writes. We have you We would have your eight is a plus. The square root off UM eight plus one should be equal to five. Well, this to mean that eight plus square root of a plus one is nine square. Tonight is three. Using the A plus three is equal to five, which means that eight plus three is an 11. So would have 11 is equal to five. That's not true, right? So we can throw out X equals eight. Would not be would not be a valid solution. So how about Access Decoder three Off X is equal to three. We would have three plus the square root of three plus one, so that would be three plus four would have three plus the square root of two plus one. So three plus the square root of four, um, should be equal to five. So three, not three times we plus right three plus this good for should be equal to five. What's good for his two? So we have three plus two. You go five, right? That's true, right? Cause two plus two is five. We have five secrets. Five. So that checks out. So therefore, X equals a three. Is it our solution? X equals eight is an extraneous solution we throw that out and get that Our solution set. Our solution set is just a set containing the number three. All right, take a

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