in-3-38-solve-each-equation-for-the-variable-check-and-write-the-solution-set-1sqrt2-xsqrt3-x1

Question

In $3-38,$ solve each equation for the variable, check, and write the solution set.
$$
1+\sqrt{2 x}=\sqrt{3 x+1}
$$

Nick Johnson · Accepted 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

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

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.

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

one plus the square root of X plus 11. So let&#x27;s go ahead. That&#x27;s first to break the ice like the square root. Let&#x27;s subtract one from both sides of the equation, so we get that X minus. One is going to be equal to the square root of X plus 11. Okay, then, to get rid of the square, root the radical. Here. We can go ahead and square both sides of the equation, but be careful with squaring X minus one so X minus one squared means X minus one times X minus one X minus one times X minus one that X minus one squared that gives us an X squared and then in minus one x minus one x his A minus two X minus one times minus one plus one ex minus one square F minus one times X minus one is X squared minus two X plus one, and then that&#x27;s gonna be equal to, well, what squaring both sides Well, that&#x27;s going to be equal to the square root of X Plus 11 squared, which just gives us That&#x27;s plus 11. So therefore we have X squared minus two X plus. One is equal to X plus 11. Okay. And now we can go ahead and get all our terms, toe one side of the equation and have a quadratic being set equal to zero. So the first go ahead and subtract X from both sides. The equation that gives us an X square and then negative two x minus X is gonna be negative. Three acts okay, and then was attracting 11 from both sides. Equation. So one minus 11 is gonna be well, I&#x27;m negative. 10 X squared, minus three X minus 10 is now equal to zero. So now we have a quadratic write in standard form that equal to zero. So can we go ahead and try and factor this quadratic? I think we can. So let&#x27;s try and factor this. This is gonna factor. I believe as well. X squared half the x Times X now 10. That&#x27;s five times two and the difference of five and two is three. Any different signs right to multiplied with negative 10 after add to a negative three extra. Also, the negative better beyond the bigger number or the five year X minus five times X plus two is equal to zero. So this would imply that X would be equal to five. Right? If X minus five is equal to zero, or or if X plus two is equal to zero, that would imply that X would be equal to negative two. Okay, let&#x27;s go ahead and check now. The solutions to make sure they&#x27;re actually solutions in our original equations, so X is equal. Five we would have five is equal 21 plus the square root of five plus 11. All right, well, five plus 11. That&#x27;s 11 12 13 14 15 60 that 16. So we have one plus two squared. 16 square 16 is for So we had that. Five is equal toe one plus four, which is true, right, because five vehicle defiance So that checks out. So therefore exp indicative five is invalid solution. How about ex being equal to negative two xnegative to we would have negative to being equal 21 plus the square root of X plus 11. Now, before I even even try to even solve this. Wait a minute. The square root of X plus 11 has to be positive, right? What could be zero, but it can never cannot. The square root of a number cannot be negative. And then we&#x27;re adding one and was somehow getting negative, too. So you can go ahead here and check out right and try toe put in negative, too. But this there&#x27;s no way that one plus the square root of a number can be equal to a negative number. So this would not check out. And therefore, um, we can disregard negative two and just say that the solution set here is the sets containing the number five. All right, take a

Problem

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

Problem 36 Hard Difficulty

In $3-38,$ solve each equation for the variable, check, and write the solution set.
$$
1+\sqrt{2 x}=\sqrt{3 x+1}
$$

Answer

$\{0,8\}$

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Algebra 2 and Trigonometry

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Algebra 2 and Trigonometry