in-3-38-solve-each-equation-for-the-variable-check-and-write-the-solution-set-xsqrtx15

Question

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

Nick Johnson · Accepted Answer

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&#x27;s go on. Let&#x27;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&#x27;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&#x27;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&#x27;t x squared and then was attracting the actual 10 x minus X that&#x27;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&#x27;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&#x27;s go ahead. And that&#x27;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&#x27;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&#x27;s good for his two? So we have three plus two. You go five, right? That&#x27;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

Nick Johnson · Answer

Okay, so we have the equation here. The square root of X plus five is equal to one plus the square root of X. Okay, well, to eliminate, eliminate. And he&#x27;s one of these great routes, right? I mean, I could just go ahead and square both sides if we do that. Squaring the left side just gives us while X plus five square root of experts five squared is just X plus five, but the right side have one plus discreet of X by square that well, that Demi&#x27;s I have one plus two squared backs times one plus the square root of X. Okay, so what does that give us? That gives us a one. But then plus what square root of X plus quarterbacks That&#x27;s plus two square roots of acts. So I still have a score the vaccine here, But then just plus X because quarterbacks times quarterbacks just pass so plus acts. So this is what I get when I square the right side. So I get X plus five is equal to well, one plus two times the square root of X plus x. Okay, well, if I combine like terms here but the exes. I can subtract off. They&#x27;re gonna be gone, right? This subtract out and they basis appear so then, while I have to square root of acts here is equal tracked The one I get two times The square root of acts is equal to off five minus one, which is four. Ah, But now I&#x27;m just go ahead and divide through by two. So we get the square root of acts is equal to 4/2, which is equal to. And now I can go ahead and square both sides to eliminate this radical right squared of X squared. It just acts so X is equal to two square which is for so you find that ecstasy could afford Is this solution valid? Wow. Go ahead and check. We have the square root of four plus five That&#x27;s the square root of nine Should be equal to while one plus the square root of four. Because access for it. Right. But this good affords two. So it&#x27;s good to nine is three of three is equal to while one plus squid A four which is too. So one plus two. Which is true, right? It goes to a straight. We have three is equal to three. So that checks out in our solution. X equals four is valid and we see that our solution set is the set containing just the number four. All right, take care.

Nick Johnson · Answer

If you have the equation here, X is equal to the square root of four X plus five. So to go ahead and to basically eliminate the square root, go ahead and square both sides of the equation. So we just get well, X square is X squared, and that&#x27;s equal to the square root of four experts, I swear, which is just equal to what&#x27;s under the radical equal to four X plus five. Okay, so then we can go ahead and against you make this a quadratic in the standard form by sending this equal to zero. So subtract all our terms to the left side. We get that X squared minus four X and then minus five is equal to zero. Okay, so we have a quadratic equal to zero. We can try and go ahead and factor this and I believe this is gonna factor So we can&#x27;t this factors as well. X squared x Times X, not five. That&#x27;s five times one right. And then, well, we need a minus four x in the middle of self. A bit of minus on the five plus on the one that&#x27;s an X squared plus one x minus five X minus four X and then minus five times one minus five. Okay, that&#x27;s factored correctly. And that is that equal to zero. So we see here that X would be equal to either five or X would be equal to negative one. Okay, but let&#x27;s go ahead. And it&#x27;s actually check these solutions to make sure they actually work in our original equation here. So exited five. Well, then we have five is equal to the square root of four times five plus five. So we have five is equal to the square root of what times five is 20 plus five is 25. So we have five is equal to the square root of 25 which is true, right? Because the sport 25 is five. Therefore, we have five is equal to five. Check now, what about if access is a negative one? But then we have negative one is equal to four times negative. One plus five. Okay, so we&#x27;d have a negative one. Negative one is equal to the square root of of four times negative. One plus 54 times negative. One plus five. Okay. Would have negative one is equal to the square. Root off four times. Negative one is negative. Four plus five. Okay, we have negative. One is equal to while the square root of negative 4.5 is positive one. So we have one could have negative one is equal to the square. Root off one. Um well, right. The square root of one is one would have native one is equal toe one and do that is not true. Right? So therefore, our solutions that here is we have a vision, extraneous solution. Our solution sets would be just to set containing five so right that as a set containing the number first, right?

Nick Johnson · Answer

four times the square root of X is equal to you. Fine. So well, let&#x27;s go ahead. That subtracted the X over to just give us busy. Isolate our term with the square root of X in it. So we do that we get four times the square root of X is equal to well, negative X plus five. Either five minus X or negative X plus five. Okay, Now, if we square both sides here, we&#x27;re gonna then eliminate the square root so four times the square root of X squared, that&#x27;s gonna be you. Go to 16 X right? And then what? We square the right side. We have two terms here, so be careful. We have to square negative five plus X meaning we have negative negative experts. Five. That&#x27;s negative. X plus five times itself. So native X plus five times negative ax plus five. OK, that&#x27;s going to equal to or native X kind of data backs Get positive. X squared. Then we have native buybacks plus negative buybacks. That&#x27;s negative. 10 acts and then plus five times five. So plus 25. So right. Negative X plus five squared is X squared minus 10 X plus 25. Therefore, we have 16 x is equal to X squared minus 10 X plus 25. Okay, let&#x27;s go ahead and we have a quadratic. Now we can go ahead and subtract the 60 next term. So then set this quadratic equal to zero. So then let&#x27;s see, we would get we have an X squared. We have my negative 10 X was attracting six x So negative 10 x minus 16 x That would be negative. 10 2026 x so minus 26 acts and then we have well plus 25 is well equal to zero. So now woken this factor. I believe it can write, um, cause this is going to be able to zero Now X squared has to be an x Times x 25 could be five times five right. But at the ad to a native 26 x, I have 25 times one right is 25 while 25 plus one after 26 in the same signs which in this case both have to be negative. And then we do add to a native 26 x for X squared minus one X minus 25 x is negatively six X and then minus 25 times one plus 25 equals zero. So we have two things being multiplied equal to zero. So at least one of those factors must be equal to zero. So therefore either X minus 25 is equal to zero, which would imply that X would be equal to 25 or X minus. One is equal to zero, which is implied that X would be equal to one. So then what? We have to go ahead. And it&#x27;s a check that both of these solutions actually work. So we check X equals one. Well, we have one plus four times the square root of one. What a screwed of one is just one who would have one plus four times one that&#x27;s one plus four. Right. We have one plus four being equal to five, which is true. We have five is he could&#x27;ve five that checks out. So therefore, X being equal toe one is a dot solution. How about ex being equal to 25 while we have 25 plus four times the square root of 25 being equal five so four times the square root of 25 would be four times five. So we would have five right? If x or not, or if I would have 25 x 25 would have 25 plus four times the square root of 25 4 times five, which is 20. We have 25 plus 20. Write those four times square with +55 is +54 times five is 2025 plus 20 would be equal to five. Is that true? No. Like I&#x27;ll be 45 5 That is not true. So you can go ahead and can throw out X equal to 25. That is an extraneous solution. So our solution set is just a set containing the number one. So you write that as you set notation here. I was set braces, So we have a set containing the number one. So there is our solution sets. All right, take care

Nick Johnson · Answer

here we have the equation three minus the square root of two experts. Five is equal to zero. So that&#x27;s first go ahead, and that&#x27;s subtract three from both sides of the equation. To get basically constant right are constants on one side of the equation. So we get we get negative the square root of two X plus five, right? If I subtract three and then we&#x27;re gonna get me equal to zero minus three, which is equal to negative three. Now, since I have the negative square roots off some stuff, you know, I can go ahead, Aiken, divide or multiply both sides of the equation here by negative one, which is going to change the signs on everything. So we get instead of negative here we get positive to get the square root off. Two acts plus five is going to be equal to a positive three. So it&#x27;s equal to three. Okay, so now in the square root of some stuff is equal to three. And go ahead and square both sides. Now the square. I have the square root of some stuff here square root of two F plus five, and I square that I just get what&#x27;s under the radical. What&#x27;s inside? What&#x27;s under the square root, right? The radical Is this the square root sign? So I square both sides of my equation, I think undo that square root, cause what&#x27;s kind of making, I guess hard, maybe to South for X here is that&#x27;s where right side, let&#x27;s go ahead and get rid of it. I just squaring both sides of the equation. So if we square both sides, we just get at two acts plus five is gonna be equal to all three squared, which is equal to nine. Okay, so now we just have a linear equation we solve for X, which we first again get are constants all to one side of the equation. So we do that first, but subtracting five from both sides, that&#x27;s gonna be leave us with two acts. Plus, they&#x27;re also just two X is equal to nine minus five, which is 56789 which is four right, nine months hours for so two X is equal Chu four. Okay, so then, while solving for X, while we just divide through by the coefficient in front of X, right to if two times actually could afford. We could go ahead and divide both sides by two, and then we&#x27;ll to over to is just equal to one that this cancels out because just getting one times X, which is acts so therefore X is equal to four divided by two, which is two. So therefore, phone that we found here that x equal to two. And therefore our solution set for this original equation is just the sets containing chew. And if we were to go and if you plug that in and check here while we have two times two, which is 44 plus five is nine square root of nine is three and three. Ministry is equal to zero. So indeed, we see that the solution set here is a set containing the number two. All right, take care.

Nick Johnson · Answer

a great year. Five plus, the cube root of a plus two is equal to three. So we subtract five from both sides. You&#x27;re gonna basically isolate our cube root. So subtracting five from both sides gives us Well, the cube root of a plus two is just gonna be equal to three minus five, which is gonna be, well, negative, too. So then, to limit the cube root, go ahead and weaken cube both sides of the equation. So the cube root of a plus two cube is just gonna be equal to well, plus two. And then that&#x27;s gonna be equal to one negative to cube. So native to cube is native to times native to time to native to case. And they have to have a negative to a positive 44 times negative two is a negative. Eat so negative to cube is gonna be equal to negative eight. So we have that A plus two is equal to negative eight so that we just go ahead and subtract two from both sides to get that a is equal to well, negative eight minus two is negative. 10. So a native 10 If you go ahead and check this well is this Doesn&#x27;t does this work while we get that five? Plus, the cube root of a plus two is equal to three, so we have five. Plus, um, the Cuba Ruud&#x27;s of well, negative 10 plus two should be equal to three. Is this true? We have five plus the cube root native 10 plus to his native eight have five plus two. Q Groot of negative eight should be equal to three. What is the cube root of negative? Eight? What&#x27;s negative to write because they give to cube is negative. Eight. So you can take the square root of a native number, But you can take the cube root of a native number because native to cube isn&#x27;t negative. Eight. So therefore, the Cube would have negative. Eight is negative. Two. We have fine plus in negative. Two is equal to three. It&#x27;s just true. Yes, by plus, and I&#x27;m gonna do something five minus two. So at five plus a native to visit with three, we have three is the coach of three, so that checks out. Therefore, A is equal to negative. 10 is a felon solution and therefore our solution set is just a set containing the only screws we found. I&#x27;m namely in negative 10. So here is our solution Set. All right, take care.

Problem

In $3-38,$ solve each equation for the variable, …

Problem 26 Hard Difficulty

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

Answer

$\{3\}$

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