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

$\left\{\frac{2}{3}, 1\right\}$

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Chapter 3

REAL NUMBERS AND RADICALS

Section 8

Solving Radical Equations

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

06:13

In $3-38,$ solve each equa…

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03:01

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here we have the equation. Two plus the square root off three X minus two is equal to three X. So we'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'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's go ahead and that we see we can factor out a common factor of three. Right? Let'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'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's go ahead, and that's try and use the quadratic formula on this inner part Reddick'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'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'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'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'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'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'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's true, right? Three is equal to three for that checks out, so therefore one is a balanced solution. How about two thirds? That's a two thirds. You would have to plus the square root of three times two thirds minus two. And that'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'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's set containing two thirds and one. All right, take a

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