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

Question

Solve each radical equation in Exercises 11–30. Check all proposed solutions.√(x + 8) - √(x - 4) = 2

Accepted Answer

Welcome back. I'm so glad you're here. We have got this radical equation and we are to solve for X. So let's rewrite that. The square root of three X plus minus the square root of five plus three X equals three. We've got two radicals in here. So what do we do? Well, we're going to go through a very similar process two times we want to isolate a radical. So get it all by itself on one side of the equal sign and we're going to square both sides to get rid of one of the radicals. We simplify everything and then again isolate the radical, get it by itself on one side and square both sides and simplify everything. So I'm going to start off by isolating that first radical the square root of three x plus 14. And I'm going to add the square to five plus three X to both sides so that we have a positive radical that we're isolating. We don't have to divide both sides by a negative. So on the left we've now got the square root of three X plus 14 equals And on the right we've got three plus the square root of five plus three X. So now as I said, we square both sides. And on the left, that is so pretty and nice because the square root of three X plus squared is just three X plus 14. And on the right, that's not so nice because we've got this factor and we're squaring it. So that means we have to multiply it by itself. That's three plus squared of five plus three X times three plus the square root of five plus three X. So now we do get to foil that out. Let's do some work on the right hand side. Our first three times three that's nine our Outsides. That's three times square to five plus three X. That's going to give us plus three square root of five plus three X. Our insides are just like that. We've got square root of five plus three X times three again. So that's again plus three squared of five plus three X. And our last we've got the square root of five plus three x times the square root of five plus three X. Or in other words we've got the square root of five plus three X squared. Now to simplify there's a couple things we can do. The square root of something squared is just going to be that number left in the middle. The square root in the squared cancel out in these two middle terms here three squared of five plus three X. Those are like terms. So we can combine them. I'll bring down the left hand side three X plus 14 equals and then bring down nine and then combine our like terms plus now six square root of five plus three X. And again that square root and the squared cancel each other out. So now just plus five plus three X. Great. Now let's bring over everything on the right hand side that's being added because again we're trying to get that radical isolated. So I'll subtract nine from both sides, subtract five from both sides and subtract three X from both sides. And look at that on the left, three X minus three X. That's 0 14 minus nine minus 50. So we got zero on the left hand side. Cool. And on the right hand side it's just that six square root five plus three X. Now we can divide both sides by six. And so on the left, zero divided by six, that's just zero equals and the sixes cancel out. So we've got the square root of five plus three X. As we did before, we're going to square bowl size to get rid of the radical. Zero squared zero equals the radicals gone now and we've got five plus three X. Let's subtract five from both sides. So on the left we now have negative five equals three X. And we'll divide both sides by three And we've got x equals negative 5/3. I'll rewrite up this up here so that when I scroll up we don't lose it. So x equals negative 5/3 was super exciting except whenever we're dealing with radicals. We need to be really careful because for our original equation, the X was always inside the radical and to bring it outside the radical to get it to a power of one which we did further down here, we finally had an X. To the power of one. But we squared it to get that power of one. And whenever we square a radical to get it to a first or second power for X. We run the risk of creating an extraneous solution. One that doesn't actually work. So we always have to check it. So I'll rewrite the original equation. The square root of three. X plus 14 minus the square root of five plus three X equals three. And now we've got to check that X equals negative five thirds. So now it's the square root of three times negative five thirds plus 14 minus the square root of five plus three times negative five thirds equals three. Well this is nice that three and the three cancel out. So it's just negative five plus 14. In the first radical minus five plus these threes cancel out negative five. Inside the other radical equals three. Okay, the left hand one. The square root of negative five plus 14. That's the square root of nine minus the square to five plus negative five. Well that's the square root of zero equals three. And the square to zero. That's just zero. So we've got the square root of nine equals three. And yeah that's true. So that means our solution for X. That we found does work in this situation and that matches answer choice. A Well done. We'll catch on the next one

Solve each radical equation in Exercises 11–30. Check all proposed solutions.√(x + 8) - √(x - 4) = 2

Key Concepts

Radical Equations

Isolating the Radical

Extraneous Solutions

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