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# Solving square-root equations: two solutions

CCSS Math: HSA.REI.A.2

## Video transcript

- [Voiceover] Let's say that we have the equation six plus three w is equal to the square root of two w plus 12 plus two w. See if you can pause the video and solve for w, and it might have more than one solution, so keep that in mind. Alright, now let's work through this together. So the first thing I'd like to do whenever I see one of these radical equations is just isolate the radical on one side of the equation. So let's subtract two w from both sides. I want to get rid of that two w from the right-hand side. I just want the radical sign. And if I subtract two w from both sides, what am I left with? Well, on the left-hand side, I am left with six plus three w minus two w. Well, three of something take away two of 'em, you're going to be left with w. Six plus w is equal to, these cancel out, we're left with the square root of two w plus 12. And to get rid of the radical, we're going to square both sides, and we've seen before that this process right over here, it's a little bit tricky, because when you're squaring a radical in a radical equation like this, and then you solve, you might find an extraneous solution. What do I mean by that? Well, we're going to get the same result whether we square this or whether we square that, because when you square a negative, it becomes a positive. But those are fundamentally two different equations. We only want the solutions that satisfy the one that doesn't have the negative there. So that's why we're going to test our solutions to make sure they're valid for our original equation. So, if we square both sides, on the left-hand side, we're going to have, well, it's gonna be w squared plus two times their product. So, two times six times w, so that's 12 w, plus six squared, 36, Is equal to, now, if you take the square root and square it, you're going to be left with two plus twelve. Now, we can subtract two w and 12 from both sides, so let's do that. So then we can get it into a standard quadratic form. So let's subtract two w from both sides and let's subtract 12 from both sides. So, subtract 12 from the right. Subtract 12 here. And once again, I just want to get rid of this on the right-hand side, and I'm going to be left with, I am going to be left with, on the left-hand side, it's gonna be w squared. See, 12 w minus two w is plus ten w, and then 36 minus twelve is plus 24, is equal to zero. Now, let's see, to solve this, let's see, is this factorable? Are there two numbers that up to ten and whose product is 24? What jumps out at me is six and four. So we can rewrite this as w plus four times w plus six is equal to zero. And so, if I have the product of two things equaling zero, well to solve this, either one or both of them could be equal zero. Zero times anything is gonna be zero. So, w plus four is equal to zero, or w plus six is equal to zero. And over here, if you subtract four from both sides, you get w is equal to negative four. Or, subtract six from both sides here. w is equal to negative six. Now let's verify that these actually are solutions to our original equation. Remember, our original equation was six. I'll rewrite it here. Our original equation was six plus three w is equal to the square root of two w plus 12 plus two w. So let's see, if w is equal to negative four, if w is equal to negative four right over, let me do something different, is equal to negative four. So, that's going to be six plus three times negative four is equal to the square root of two times negative four plus 12 plus two times negative four. So this would be, this is, negative twelve here. This is negative eight here. This is negative eight here, so you have six plus negative twelve, which is negative six, is equal to the square root of negative eight plus 12, is four, plus negative eight. So that would be negative six is equal to two plus negative eight, which is absolutely true. So this is definitely a solution. And let's try w is equal to negative six. So, we is equal to negative six. So, we're gonna get, if we look up here, we're going to have six plus three times negative six is equal to the square root of two times negative six plus 12 plus two w. So, this is gonna be negative 18. This is going to be negative 12. This is negative 12. Negative 12 plus 12 is zero. Square root of zero, this is always zero. And then two times, and actually let me, I shouldn't have written a w there, I should have written two times negative six. So back to what I was doing, this right over here is negative eighteen. This is two times negative six plus 12. This is all zero. Square root of zero is zero. And then this is negative twelve. So you get six plus negative of eighteen which is negative 12. Is equal to zero plus negative 12. Negative 12. Which is absolutely correct. So these are actually both solutions to our original radical equation.