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

$\{0,8\}$

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Chapter 3

REAL NUMBERS AND RADICALS

Section 8

Solving Radical Equations

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

00:57

In $3-38,$ solve each equa…

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03:59

03:39

04:23

05:57

06:13

02:28

00:35

02:48

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'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'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's go ahead. And that'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.

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