in-3-38-solve-each-equation-for-the-variable-check-and-write-the-solution-set-sqrt3x2

Question

In $3-38,$ solve each equation for the variable, check, and write the solution set.
$$
\sqrt[3]{x}=2
$$

Martha Richards · Accepted Answer

We&#x27;re gonna solve the solution here Checker answer and then provide the solution set for X. So here, when I have this cute, rude, isolated Aiken, just Q both sides. So then the left becomes X and two cubed is eight. So we&#x27;re gonna check her answer here. So when I put this back in the original equation, cubed root of eight is too. So this checks out. So our solution set for this equation here is for when acts is equal to eight.

Nick Johnson · Answer

here we have the equation. Two plus the square root off three X minus two is equal to three X. So we&#x27;re gonna do is first subtract the two over. Just isolate the square root here. Square three X minus two That gives us the square root of three X minus two is equal to three X well minus two. So if we square both sides, we then eliminate the square root so we get while three x minus two squared It just takes minus two is equal to three X minus two squared but three X minus two squared is three x minus two times three x minus two which gives us well a three x times three x is nine x squared and then a minus six x minus six X is a minus 12 x and then plus four. So three of minus two squared is nine x squared, minus 12 X plus four. So therefore, we square the left side. We just get three X minus two. Many square the right side. We get nine X squared minus 12 ax plus four. So then we take our quadratic Here we subtract the three X term and try and add to both sides to set our quadratic equal to zero. So we get nine X squared minus 15 x, because minus 12 x minus three X is minus 15 x and then we&#x27;re adding to to both sides. So four plus two or six So get nine X square minus 15 X plus X is equal to zero. And we could try and back to this, and we might come into sexism. Well, first, let&#x27;s go ahead and that we see we can factor out a common factor of three. Right? Let&#x27;s pull out a common factor of three. So we get three times while three X squared gives us in nine X squared and in three times minus five. X gives us a negative 15 x and in three times to use a six. So plus two o k. Is equal to, um, zero, right? That&#x27;s nine X squared minus 15 X plus 60 0 Now we could try and maybe trying factor this so we might run into some problems. Elves. Let&#x27;s go ahead, and that&#x27;s try and use the quadratic formula on this inner part Reddick&#x27;s we Now we know that either is a +300 which streets never equal to zero. So have even need that. We need three X squared, minus five acts plus two to be equal to zero. But you can go ahead and use the quadratic formula to solve for X, so we know that X is equal to or negative. B. Zaheer B is indicative. Five. So we have negative negative five. We have five plus or minus the square root of B squared native. Five squared. That&#x27;s negative. Five. Squared um, so native B plus or minus the square root of B squared minus four times a time. Sees or four times a his three I&#x27;m see, which is to all divided by +28 divided by two Times Street. OK, so this gives us while five plus or minus the square root of what while native five square is 25 square root of 25 minus four times three times to four times three is 12. 12 times two is 24 25 minus 24 all over six. Okay, three times to a six, but 25 miles. 24. That&#x27;s one. We have five plus or minus the square root of one, which is just one right, so we have five plus or minus one all over six. So therefore, Acts writes X is equal to five plus or minus 1/6. Therefore, X is either equal to while five um plus 1/6. That&#x27;s 6/6, but six over sex is just even though one for exercise with one or exercise equal to five, minus 1/6 or five minus one over sex. That&#x27;s four over sex, right? But for over 65 both numbers by to hear and forward six is equal to 2/3, which two thirds so therefore X is equal to either one or two thirds. Now go ahead and check both the solutions and make sure they&#x27;re actually valid solutions to our original equation. OK, way up here, X is equal to one. Well, let me just get two plus, while the square root of three times one, which is just three minus two, should be equal to three times one or three. So therefore, a to be equal to with three minus two is just one screwed of one, just one, so you have two plus one is equal to three. That&#x27;s true, right? Three is equal to three for that checks out, so therefore one is a balanced solution. How about two thirds? That&#x27;s a two thirds. You would have to plus the square root of three times two thirds minus two. And that&#x27;s what the equal to three times two thirds okay or three times two thirds is just equal to write because, such as 3/1 threes cancel out. Just get two or two minus two is equal to zero. So we get that two plus two squared 00 So that, too, is equal to while three times two thirds is also equal to write because can three is cancel out become one and the the two is equal to which checks out. That&#x27;s also true. So therefore, both solutions here are valid. X is equal to one or act is equal to two thirds. So therefore, our solution set is the second can be these two values one and two thirds or two thirds and one remember order you put him in so we have the solution set. Here it&#x27;s set containing two thirds and one. All right, take a

Nick Johnson · Answer

three plus the square of four x minus street is equal to two X, so go ahead. And to isolate the radical term here, that thing I knew need the square root. We can go ahead and subtract three X from both sides Equation that&#x27;s gonna give us Well, the square root of four X minus three is then going to be equal to out two X minus street. Okay, then we could go ahead and square both sides of the equation because the square root of four X minus three squared is just gonna give us what&#x27;s under the radical is gonna give us for X minus three, but squaring two x ministry with a square. We have two terms here, so we&#x27;re squaring two X minus three. So that means we take two X minus three times two X minus three and we have to distribute here. So this is gonna give us Wow, This is equal to two x times two x That&#x27;s four x squared and get two x times negative. Three such a negative six x and then well minus another six x so minus six x minus six X is gonna B minus 12 acts and then we have minus three times minus three is gonna be a plus night. So two X minus three squared is four X squared minus 12 X plus nine. So therefore, if we square about side or the left side, we just get again. Which underneath the square root So the square root of four X minus three squared is just equal to four X minus three. And then that&#x27;s equal to what two X minus three squared, which we see here is four X squared, minus 12 acts plus nine. Okay, now we can go ahead and weaken. Basically, subtract these two terms to the other side to then have a quadratic which is equal to zero. So we&#x27;re gonna have Well, we we have a four x squared term. So I have four x squared. But then we have native 12 x subtracting for X so minus 12 exit minus four x Well, that&#x27;s 12 13 14 15 16 minus 60. Next, we have four x squared, minus 16 acts, and then we have while nine and then we&#x27;re adding three to both sides. So nine plus three is 10. So we have four x squared minus 60. Next 10 is 12 right? Nine plus three is 12. So nine months we get plus 12. Right? Adding three. The both sides for nine plus tree is 9, 11 12 guests trouble. So now we have this being equal to zero. Okay, so? Well, um, well, we can definitely factor out a four year, right, because four goes into four. X squared goes in 16 and goes into 12 by factor out a four. We have four times Well x squared, and then four times on minus four. X gives us in 1916 x and then four plus three is 12 7 plus three. Okay. And that&#x27;s equal to zero. Okay, so now I can go ahead and factor. Um, the inside here is a quadratic. So we have four times trying. Factor this quadratic. So that&#x27;s gonna factor as where x squared is x times x three. That&#x27;s three times one. I need the same signs right at the edge with negative for X off. Both of these are negative, right? That seem to be an X squared minus one x minus three x minus four X plus three K is now equal to zero So now I have, well, the product of what? Actually three things being equal is there. Also, at least one of the factors have to be zero or four is never zero right. So therefore either X minus 30 or X minus 10 X minus three is zero. That implies that X is he, won&#x27;t you? Three or if x minus 10 that implies that X is equal to one. Okay, so we have two solutions we found. But now let&#x27;s go ahead. And that&#x27;s check and make sure that both of these solutions are actually valid solutions. So if we check X equals three, we have three plus the square root of four times three minus streets. So that would be, um they were checking checking over here now. So we have three plus the square root of while four times we are checking. Executed 34 times three is 12 so 12 minus three. That should be equal to two X. So the two times three, which we go to six. Is this true? All we have three plus The square root of 12 minor street or 12 minus three is 12 11 10 9 the square root of nine. Is he gonna three? We have three. Plus three is equal to sex, which is true, but we&#x27;re six. A secret sex, therefore, that checks out and therefore X is equal to three is a valid solution. How about X? Be equal till one X is equal. No one would have three plus the square root of while four times one minus street. There will be four, Um, while four minus three. Which is right. Rex is one. Yet four times one is four for minus three is one square root of one is one for the three. Plus the square root of one is equal. Chew. Um, two times. Times one. Okay. Is this true? Well, it&#x27;s good of one is one you have three plus one would be equal to two. That would mean that four is equal to two, and that&#x27;s not true. So we go ahead and we throw out X is equal to one, and we see that the only right that we have an extraneous solution X equals one. The only valid solution is X equals three. So therefore, our solution set would be the set to make a set brackets here. So the set containing Justin number three there is our solution set. All right, take it.

Martha Richards · Answer

you were asked to solve this equation, check for it and then provide the solutions that so when the radical is isolated, we can just square both sides. So I get one plus X on the left is equal to nine, cause three squared is nine. And then I subtract one on both sides, and I end up here with X equal to eight. So now let&#x27;s check this in the original equation. So I replaced acts with eight. And here, when I have eight plus one, that&#x27;s the square root of nine and the spirit of nine is three. So this does check out. So our solutions that for the equation above here is for X equal to eight.

Nick Johnson · Answer

having equation here. One plus the square root of X is equal to three. So on the left side of the equation, I have two terms, right? I have one in the square Durex. So I have an ex term and have a constant. Let&#x27;s go ahead and move. All are constants toe one side of the equation by just subtracting one from both sides. Track one from both sides. One minus 10 I just get zero plus bread of expert justice. Square root of acts is equal to three minus one squared of access equal to two and then the south for X because I have the square root of X Aiken Square, both sides to undo the square root is the square root of X squared is equal to X, and then we square both sides so excessively could the two square or two times two, which is equal to four so X secretive form, meaning that our solution set contains just the number of four. There&#x27;s our solutions said All 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

Problem

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

Problem 30 Easy Difficulty

In $3-38,$ solve each equation for the variable, check, and write the solution set.
$$
\sqrt[3]{x}=2
$$

Answer

$\{8\}$

Related Courses

Algebra 2 and Trigonometry

Related Topics

Discussion

Top Educators

Recommended Videos

Watch More Solved Questions in Chapter 3

Video Transcript

Ann Xavier Gantert

Algebra 2 and Trigonometry

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$\{8\}$

Algebra 2 and Trigonometry

Ann Xavier GantertAlgebra 2 and Trigonometry

Ann Xavier Gantert

Algebra 2 and Trigonometry