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Studying for a test? Prepare with these 5 lessons on Radical equations and functions.
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We're asked to solve for x. So we have the square root of the entire quantity 5x squared minus 8 is equal to 2x. Now we already have an expression under a radical isolated, so the easiest first step here is really just to square both sides of this equation. So let's just square both sides of that equation. Now the left-hand side, if you square it, the square root of 5x squared minus 8 squared is going to be 5x squared minus 8. This is 5x squared minus 8. And then the right-hand side, 2x squared is the same thing as 2 squared times x squared or 4x squared. Now we have a quadratic. Now let's see what we can do to maybe simplify this a little bit more. Well, we could subtract 4x squared. Or actually, even better, let's subtract 5x squared from both sides so that we just have all our x terms on the right-hand side. So let's subtract 5x squared from both sides. Subtract 5x squared from both sides of the equation. The left-hand side, this cancels out. That was the whole point. We're just left with negative 8 is equal to 4x squared minus 5x squared, that's negative 1x squared. Or we could just write negative x squared, just like that. And then we could multiply both sides of this equation by negative 1. That'll make it into positive 8. Or I could divide by negative 1, however you want to view it. Negative 1 times that times negative 1. So we get positive 8 is equal to x squared. And now we could take the square root of both sides of this equation. So let's take the square root of both sides of this equation. The principal square root of both sides of this equation. And what do we get? We get, on the right-hand side, x is equal to the square root of 8. And 8 can be rewritten as 2 times 4. And this can be rewritten as the square root of 2 times the square root of 4 is equal to x. I don't like this green color so much. And what's the square root of 4, the principal root of 4? It's 2. So that right there is 2. So this side becomes 2, this 2, times the square root of 2. And that is equal to x. Now let's verify that this is the solution to our original equation. So let's substitute this in, first to the left-hand side. So on the left-hand side, we have 5 times 2 square roots of 2 squared minus 8. And then we're going to have to take the square root of that whole thing. So this is going to be equal to-- we're just focused on the left-hand side right now. This is equal to the square root of 5 times 2 squared, which is 4, times the square root of 2 squared, which is 2. And then minus 8. And this is 5 times 4 is 20 times 2 is 40. And then you have 40 minus 8 is 32. So this is equal to the square root of 32. Square root of 32 is the same thing as the square root of 16 times 2. The square root of 16 is 4. So this is the same thing as the square root of 16 times the square root of 2. Or 4 square roots of 2. So that's what the left hand simplifies to when we-- and remember, the original equation didn't have these squares here, so if you just look at the green part, the green part on the left-hand side just simplified to 4 roots of 2. Let's see what 2x simplifies to. Our original right-hand side was just the 2x. That's parentheses with the square added later. So what's 2x? 2 times 2 roots of 2. 2 square root of 2. Well that's just 4 square root of 2. So when x is equal to 2 square roots of 2, the left-hand side equals 4 square roots of 2. And remember, the left-hand side looked like this when we started off. The left-hand side when we started off didn't have that there. I want to make that clear. So when you substitute this back into this left-hand side, you get 4 square roots of 2. When you substitute it back into the original right-hand side, you get 4 square roots of 2. So it's definitely the right-- I'm trying to write in black. It's definitely the right solution.