in-3-38-solve-each-equation-for-the-variable-check-and-write-the-solution-set-1sqrtx3

Question

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

Nick Johnson · Accepted 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,

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

plus three is equal to the square root of one minus three X. So we&#x27;re gonna go ahead with a square both sides of the equation to get rid of the radical but squaring expose three. Let&#x27;s expose Story Square is exposed three times X plus three. So squaring left side we get X plus three times X plus three Okay, that&#x27;s gonna be equal to an X squared plus three x plus three X plus six acts and then plus three times three plus nine. So x squared plus six x plus nine is X plus three squared and that&#x27;s gonna be equal to while the square root of one minus three X squared just gives us what&#x27;s under the radical, namely one minus three acts. Okay, so now we&#x27;re gonna go. We&#x27;re gonna go in, um, get all our terms on one side of the equation, but have a quadratic equal to zero. So first, let&#x27;s go ahead and add three extra both sides. So we get X squared. Plus, while six x plus three x is plus nine acts and then we&#x27;re going to subtract one from both sides, so nine minus one is eight to get plus eight is gonna be than equal to zero so that we can We have a quadratic here in standard form setting with a zero. But you can go ahead and try and factor Mr Factor as, um, x times X is x squared it. Well, that&#x27;s eight times one, right. We need everything positive. Here. Is that add to a nine X in the middle. Okay, this is equal to zero. This means that either X is equal to or negative eight or access is equal to negative one. All right, let&#x27;s go ahead and out. Chuck these solutions and make sure they work in our original equation. The first score headed check, maybe exit was negative. One X equals negative one. We get negative one plus three. So native one plus three is equal to we have to is equal to the square root of one minus three times a negative one. So we have two is equal to the square root of one minus three times a negative one. Um, is a negative three. If you have one minus and negative three. That&#x27;s one plus three was just four. You have to is equal to the square root of four, which is equal to two. Right? We have two is equal to four is to have two is equal to two, Which checks out. So therefore, um negative one is a solution. How about negative? Eat? If X is negative eight, we have negative eight plus three. That would be 345678 A. B negative five would have Negative five is equal to the square root of one minus three times. Um A Did you did you do three times a negative, right? Which are gonna get a great so three times a negative. It&#x27;s all right. You have negative. Five is equal to the square. Root off while one minus three times a debate is a negative. 24. We have one minus and native 24. That&#x27;s one plus 24 which is 25. If you have negative five is equal to the square root of 25. Is that true? Well, we have negative five right is equal. Choose grew. 25 is well five. So negative five is equal to five is not true. Everything go ahead and can throw out X equals negative eight and therefore our solution set is just a set containing negative one. So we get, um the solution set here is the set containing negative one. All right, take her.

Nick Johnson · Answer

So here we have one plus the square root of two acts is equal to the square root of three X plus one. So we tempted here to well square both sides. So if we do so what happens? Why square the left side have two terms I don&#x27;t just, you know, limits the square root because I have one plus two squared two x I square that I get one plus the square root of two x right times itself. So times one plus the square root of two x What is this sequel that equals one times One which is one and I get well I get plus the square to two x plus the square directs that plus two times the square root of two acts Okay, And then while plus squared rex times where tracks is just Rex So plus two x Okay, So this is what I give my square. The left side screwing the right side just eliminates tyrannical Because the square root of three experts one square is just the experts one. So here, if we get if we square both sides, we get while one plus two times the square root of two acts plus, Truax is equal to three X plus one. Okay, so if we go ahead and we combine our likes terms here where we all get well, I can subtract that. See the two X here, the ones right, This cancel our ex, attract one from both sides and they&#x27;re gone. And then I get well, I get two times the square root of two X is going to be equal to well, three X minus two X is just one act which is equal to X. Okay, so well, now, Okay, so I have two times the square root of two accessible tax. If I divide through by two, I get the square root of to axe is equal to X over to. And now I can go ahead and square both sides to then eliminate this radical by square both sides. The square two X squared is just two X is that equal to X? Over two squared is gonna be well X squared over four now, while I can multiply both sides by four to get rid of this fraction. So I multiplied by four. You get four times two x is gonna be eight axe is gonna be equal to X. Um, squared. Okay. Well, now we can basically set this thing as a quadratic. So we have X squared minus eight. X is equal to zero. Okay, being a factor out, a common factor of acts. So we have X Times X minus eight is equal to zero. And now we see that well, we have two factors being marked by together equals zero. So that means that either X is equal to zero or four X is equal to eight. Okay, so now let&#x27;s go ahead. And that&#x27;s check the solutions to make sure they are valid solutions. So X is equal to zero. We have one plus two square to come. Zero for them. Zero is zero scored. A 00 is one plus zero, which is one on the left side and the right side. We get well, the square root of three times zero plus 10 plus one, which is this crowd of one which is one to get that one is equal to one which checks, right. So therefore, um X. Being equal to zero is a ballot solution. How are extinct equals eight. Well, we get one plus the square root of four times to get one plus the square root of what times four times eight. Excuse me is 16. So get one plus the square root of 16 which is Wall Street. 15 is four want us for which is five. So the left side of the equation is five and the roadside or the square root of three kinds eight plus 13 times eight is 24 24 plus one is 25 the square root of 25 on the right side, which is equal to five. So we get the five equal time, which also checks out. So therefore, both of these solutions are valid. X is equal to zero or Xs equals eight. When you see that our solution set is the set containing the two numbers, namely zero and eight. All right.

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.

Nick Johnson · Answer

here We have immigration. The square root of X plus four minus the square root of X minus one is equal to one. So well, if we were to Hmm. Let&#x27;s see. How about if we add, um, a square root of X minus one to both sides, So that would then give us the square root of X plus four within the equal to well, one plus the square root of X minus one. Why do we do this? But now we go ahead and square both sides. Square in the left side is just going to give us X plus four, right? The squarely of plus four square is just equal to X plus four. So but the right side we have what? One plus the square root of X minus one. So we square the right side we get while we get one plus the square root of X minus one times, right. One plus the square root of X minus one times itself. But the needs to square it Well, this was gonna equals one times. One is one. Then we get one time squid of X minus one stashed a squirt of X minus one plus right, Um, plus another one times the square X minus one that&#x27;s gonna give us while plus two times the square root of X minus one on Ben while X minus one times X minus one just gives us X minus one so that we get while plus, um plus X minus one. So plus X minus one. Okay, so squaring the left side again, the squid of X plus four squared Just could just x plus four. So we get X Plus four is gonna be equal to again squaring This gives us this year&#x27;s We get one plus two times the square root of X minus one, uh, plus X minus one. Okay, well, the exes are gonna cancel out right track action. Both sides and the exes are gone. Um, And then, well, we can go ahead and we can add one to both sides. Um, well, actually, not really. Because you have a one minus one. So that&#x27;s gonna have to cancel. It becomes zero. Okay, so let me get well, just a four is equal to two times the square root of X minus one, right. That&#x27;s what this is gonna basically simplify down too. And then we can go ahead and divide through by two to get for over two or just two is equal to the square root of X minus one. And now we can go ahead and square both sides to eliminate this square root, so squaring both sides now gives us while two square is four is equal to just X minus one. So adding one the both sides gives us that X right is equal to five. So we have that X is equal to fight, right? Just adding one to both sides. You have exercise five. Now go ahead and check the solution in our original equation. Here is this true or the square root of five plus four is just the square root of nine and then we have minus, while the square root of five minus one that&#x27;s the square root of four that should be equal toe won with a score of nine is three, and a squirt of four is to just have three minus two, which indeed is equal to one writes that simplifies down to one. Is it with the one of hat is true that checks out and therefore our solution here. X equals five is a balance solution in our solution set is the set containing just the number of five. All right, take care.

Problem

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

Problem 9 Easy Difficulty

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

Answer

$\{4\}$

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

$\{4\}$

Algebra 2 and Trigonometry

Ann Xavier GantertAlgebra 2 and Trigonometry

Ann Xavier Gantert

Algebra 2 and Trigonometry