Solve each radical equation in Exercises 11–30. Check all prop...

Question

Solve each radical equation in Exercises 11–30. Check all proposed solutions. $\sqrt{1 + 4 \sqrt{x}} = 1 + \sqrt{x}$

Accepted Answer

Welcome back. I'm so glad you're here. We have this equation, the square root of 18 square root X plus nine on the left hand side equal to two squared X plus three. And we are to solve for X. Well, how do we do that with all these radicals? Well, first let's get rid of this big radical on the left hand side by squaring it. And what we do to one side we need to do to the other. So the square root of 18 square root X plus nine squared Well that's just 18 squared X plus nine equals And now the right hand side, we do need to take those two factors two squared X plus three times two squared X plus three and foil those out. So our first two square root X plus three, pardon me. And so our first are going to be two squared X. Times two square root X. That's going to be four square root X squared. Our Outsides will be two squared X. Times three, which will give us a plus six squared X. Our insides give us three times two square root X. Which will give us another plus six square root X. And our lasts will give us a three times three. That's nine. So plus nine and will bring down the left hand side 18 squared of X plus nine. Right now, we can simplify a couple things on the right hand side. The squared of X squared. Well, that's just going to be X. And we can combine the like terms six squared X plus six squared X. So let's do that will bring down on the left 18 squared X plus nine equals no longer four squared X squared but now four X plus and six squared X plus six square root X. Is 12 square root X plus nine. Now let's bring the left hand side of the equation over to the right hand so that we can keep the number that has the largest degree which in this case is four X will keep that positive. It'll be a little bit easier to factor. So on the right hand side or the left hand side of the equation. Now we have zero equals and bringing down on the right four X, 12 squared x minus 18 squared X is negative six square root X and nine minus nine. That cancels out that zero. Great. Now we look at the equation and we see that everything is divisible by two. So zero divided by two. That's still zero equals four X divided by two. That is two X and minus six X. Or six square root X divided by two is three square root X. Well, now, what we can do is we can see if we can factor anything out of the two terms on the right hand side and we can factor out a square root of X. So zero equals squared of X times. And what is two X divided by the square root of X. Well, we know that we could rewrite the squared of X as X to the one half and two X in the numerator. Well that has a degree of one. And when we are dividing our um variables that have the same base, we're going to subtract their exponents. So it'll be X to the first minus one half which will give us two X to the one half. And as we just said, we can rewrite X to the one half square root of X. So inside of our parentheses here, this will be two squared of x minus and three squared of X divided by squared of X. Well the squared of X is cancel out so it's just minus three here. Great. We've got two factors and we can set both of them equal to 0 to solve. So let's start off with squared of X equal to zero. And in order to get that X by itself we're going to square both sides. So the squared of X squared, that is just going to be X and zero squared. That's just zero. So one of our exes is zero. Now for the other one will set two squared of x minus three equal to zero. Let's bring the three to the other side, add three to both sides. So we have two squared of X equals three. And now we will divide both sides by two. So on the left hand side it's just the squared of X. And on the right hand side it's now three halves to get rid of that square root. We are going to square both sides. And on the left it is just X. And on the right. Well we're going to take three over two squared which is the same as three squared over two squared. Three squared is nine and 22 to the fourth and two squared is four. So our other answer for X. X equals 9/4. Great, wonderful. We have an answer and now we are asked to check the solutions. So I'm going to rewrite what that equation is here at the bottom and then scroll down for a little bit more room. So we've got the square root of 18 square root X plus nine equals two times the square root of X plus three. And anywhere we have an X. We're going to substitute first nine forts and see what we get and then the other one will substitute a zero. So for the first one X equals 9/4 will have the square root of 18 times the square root of 9/4 plus nine equals two times the square root of 9/4 plus three. Well, what is the square root of 9/4? We know that's the same as the square root of nine over the square root of four which is 3/2. So simplifying this, we've got the square root of 18 times three halves plus nine equals two times three halves plus three. And continuing to simplify while 18 times three halves, the 18 and the two are both divisible by two. So that'll give us a nine times three. And that'll be the square root of 27 plus nine equals and two times 3/2. Will those two's cancel out? And we've got three plus three. So the square root of 27 plus nine? Well that's the square root of 36 on the right hand side, three plus three, that's six. And yeah, the square root of 36 is six. That is correct. So X equals 9/4 does work. Now let's go ahead and substitute a zero in for everywhere we have an X. In our original equation. So the square root of 18 times the square root of zero plus nine equals two times the square root of zero plus three. And well this is great because the square to zero, that's zero. So we've got 18 the squared of 18 times zero plus nine equals two times zero plus three. Well 18 times zero, that's zero. So we've just got the square root of nine on the left hand side and two times zero, that's zero. So we've just got three on the right hand side. And yes the squared of nine is three. So that one checks out on the right hand side. We know that both of these answers work X does equal 9/4 and X does equal zero. So we scroll way back up and we take a look at our answer choices and that matches with answer choice. A well done, we'll catch you on the next one.

Solve each radical equation in Exercises 11–30. Check all proposed solutions. $\sqrt{1 + 4 \sqrt{x}} = 1 + \sqrt{x}$

Key Concepts

Radical Equations

Nested Radicals

Extraneous Solutions and Verification

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Solve each radical equation in Exercises 11–30. Check all proposed solutions. 1+4x=1+x\sqrt{1 + 4\sqrt{x}} = 1 + \sqrt{x}

Key Concepts

Radical Equations

Nested Radicals

Extraneous Solutions and Verification

Watch next

Solve each radical equation in Exercises 11–30. Check all proposed solutions. $\sqrt{1 + 4 \sqrt{x}} = 1 + \sqrt{x}$