in-3-38-solve-each-equation-for-the-variable-check-and-write-the-solution-set-b-3sqrt3-b-11

Question

In $3-38,$ solve each equation for the variable, check, and write the solution set.
$$
b-3=\sqrt{3 b-11}
$$

Nick Johnson · Accepted Answer

the equation here, B minus three is equal to the square root of three B minus 11. We can go ahead and in square both sides the equation. So to get rid of this radical right, so we have B minus three squared or B minus three squared is B minus three times B minus three. So that&#x27;s gonna be equal to be square. And then we have a minus three B minus three beats that&#x27;s minus six B and then minus three times minus three is a positive nine. So be ministry squared is B squared minus six B plus nine that&#x27;s gonna be equal to the square root of three B minus 11 squared. Which is just gonna give us what&#x27;s under the radical. Give us just three B minus 11. All right, so now let&#x27;s go ahead. And that&#x27;s while move all our terms to one side of the equation would set a quadratic equal to zero. So it was attract the three B term. First, we&#x27;re gonna subtract three big from both sides. Okay, so we still have a B squared on left side, and then we have on negative six B minus three beats. That&#x27;s gonna be a negative nine b and then we&#x27;re gonna add the 11. Right? We have three B minus 11. So when I add the 11 to both sides, we have 11 plus nine. What is that? That&#x27;s while 10 plus nine is 19 plus the one more that&#x27;s gonna be 20 b squared, minus nine B plus 20. And now this is gonna be equal to zero. So now we can go ahead and try and factor this quadratic we have here. So B squared minus nine B plus 20 is going toe factor as well. We have a B and B for B squared. 20 is gonna be while five times four right and five plus four is nine. So five and four should work here and we need opposite signs when he&#x27;s running same size, same signs the ad to a on positive 20 but they haven&#x27;t add to a negative nine. So both of these are gonna be negative. And there&#x27;s just gonna be equal to zero. So therefore, if we have B minus five times, B minus four equals zero. The product of two things being equal to zero. At least one of the factors must be equal to zero. So either B minus five is equal to zero, which implies that B is equal to five or B minus four is equal to zero. That implies that B is equal to four. So either because you got a five or be is equal to four. Okay, let&#x27;s go ahead and check both of these solutions and mix. Do they actually work in our original equation? So be is five. Well, then we have five minus three, which is gonna be equal to two. Right? And that&#x27;s gonna be equal to the square root of well, three times times five minus 11. We have two is equal to the square root of five times because 15 of minus 11 is, uh, 432 is equal to describe it. A four which is true, right? Because it affords to we have to is equal to two that checks out. So five is a solution. How about four? Well, B is four. We have four minus three, which is equal to one, and then that&#x27;s supposed to be equal to the square root of three B minus 11. So three times four times square root of three times four minus 11. So we have the square root of three times four minus 11 while that&#x27;s becomes for his 12. The square root of 12 minus 11 which is one we have one, is equal to the square root of one, which is true, right? Because want this sort of one is one of one is equal to one. So that checks out. So we see that both solutions here five and four work out perfectly. Therefore, our solution set would be the set containing these two values, namely five and four. So there&#x27;s a right that in set notation. That&#x27;s the set containing five and four. All right, take care.

Martha Richards · Answer

Okay, so we&#x27;re going to solve this equation for be checker answers and then provide a solution set so unloved here we have through my speed, and we just square that because we&#x27;ll square this side with the isolated radical. So I get when I do three minus b square, and I&#x27;ve got a foil that out. So three times three is nine of three times of minus. B is minus three B, and then another minus three gets B minus 60 and then a negative b times a negative. B is positive b squared. And so on the left, I&#x27;m going to just rearrange this here and then on the right. I get the three B minus 11 to get all of it over to the one side. I&#x27;m going to subtract three b and at 11 to both sides on the right. I&#x27;ll get zero, cause all of that cancels up on the left. I&#x27;ll get it B squared. A negative sex minus of three gets me a negative 90 plus 20. Now, when I factor this out, I get a B for the first term, and then I need in order to get a positive here and a negative for the second term. I&#x27;ve gotta have two negatives, and then the factors of 20 that add up to nine are four and five. So then solving for each of these I get that B is equal to four and B is equal to five. So let&#x27;s test both of these out. In the original equation, I have three minus four, equal to the square root of three times four minus 11. And so three minus four is negative one and then I get a 12. Minus 11 is the square root of one, and a negative one is not equal to one. So let&#x27;s also try this other number here, this five. So if I have three minus five, equal to the square root of three times five minus 11 I get three. Minus five is a negative two and then a five times of three 15. Minus 11 is the square root of four, and the negative, too, is not equal to two. So both of these do not work. And so the solution sat here is no

Nick Johnson · Answer

you have the equation here. Two plus the Q groot of three B minus two is equal to six. That&#x27;s first, and let&#x27;s go and subtract two from both sides of the equation to just isolate our cube root. So subtracting two from both sides just leaves the while the cube root off three B minus two is gonna be equal to six minus two, which is equal to on four. Right? Okay, then, while we can go ahead and we can cube both sides of the equation here too, then eliminate the cube root. So if we if we cube if we cube the left side what we get me a three beat the cube root of three B minus four Cube is just gonna be equal to three B minus two. And then if we cube the right side, get four cube. So that is four times four times four, which is 64 right, four times for 16 time. Forget 64. So get three B minus two is quote shoes 64. Okay, so I just have a linear equation. Weaken go in ad to Tebow size to isolate RP term. So we have the three B is equal to 64 plus two over adding to the both sides here. That&#x27;s equal to 66. And then we just go ahead and divide through by three. Right? Just be over. Three is one that is going to be is equal to 66/3 and 66. Divided by three is equal to 22. So that be is equal to 22. Does that check out while B is 22? We have two plus. The cube root of three times 20 to minus two should be equal to six. So we have not two times. Two plus me. Right. So, um, is this true? Well, we get two plus the cube root of three times 22. So that&#x27;s, um, 66. Right? Minus two is 64 the cube root of 64 which is for right. So we get that two plus again. The Cuba of 54 years forced me. A two plus four is equal to six. Is that true? Yes, you&#x27;ve got six is equal to six, so it checks out and we get that be equal. 22 is about a solution. So therefore, our solution set is the set containing 22. All right,

Martha Richards · Answer

were asked to solve this equation checker answer and provide the solution set. So here, when I have the radical isolated, I could just wear both sides. I get a to be on the left, eight square to 64. I divide both sides by two to solve for B, and I get that is equal to three 32. So let&#x27;s put this into our original equation and check to make sure it&#x27;s right. So I have two times 32 is equal to eight square root of that, and then I have. The square root of 64 is equal to eight in the square to 64 is eight, so this checks out, so our solution set for the equation above is for be equal to 32.

Nick Johnson · Answer

have two times The square root of B is equal to eight. So we go ahead and we divide through by two. We divide both sides of the equation by to let me just get the square, Toby. Because to over two is one that cancels out Just get the square root of B is equal. Judit divided by two which is equal to four. Okay, so the square to be is he gonna four? That means that I would just square both sides We get that b is they go to foursquare or four times for which is 16. So here our solution set is the set containing 16 So here would be our solution set. All right, take your

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 23 Hard Difficulty

In $3-38,$ solve each equation for the variable, check, and write the solution set.
$$
b-3=\sqrt{3 b-11}
$$

Answer

$\{4,5\}$

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$\{4,5\}$

Algebra 2 and Trigonometry

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Ann Xavier Gantert

Algebra 2 and Trigonometry