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We're asked to solve for x.

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So we have the square root of the entire quantity 5x squared

4
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minus 8 is equal to 2x.

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Now we already have an expression under a radical

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isolated, so the easiest first step here is really just to

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square both sides of this equation.

8
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So let's just square both sides of that equation.

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Now the left-hand side, if you square it, the square root of

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5x squared minus 8 squared is going to be 5x

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squared minus 8.

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This is 5x squared minus 8.

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And then the right-hand side, 2x squared is the same thing

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as 2 squared times x squared or 4x squared.

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Now we have a quadratic.

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Now let's see what we can do to maybe simplify this a

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little bit more.

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Well, we could subtract 4x squared.

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Or actually, even better, let's subtract 5x squared from

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both sides so that we just have all our x terms on the

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right-hand side.

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So let's subtract 5x squared from both sides.

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Subtract 5x squared from both sides of the equation.

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The left-hand side, this cancels out.

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That was the whole point.

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We're just left with negative 8 is equal to 4x squared minus

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5x squared, that's negative 1x squared.

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Or we could just write negative x

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squared, just like that.

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And then we could multiply both sides of this equation by

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negative 1.

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That'll make it into positive 8.

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Or I could divide by negative 1, however you

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want to view it.

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Negative 1 times that times negative 1.

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So we get positive 8 is equal to x squared.

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And now we could take the square root of both sides of

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this equation.

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So let's take the square root of both

40
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sides of this equation.

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The principal square root of both sides of this equation.

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And what do we get?

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We get, on the right-hand side, x is equal to the

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square root of 8.

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And 8 can be rewritten as 2 times 4.

46
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And this can be rewritten as the square root of 2 times the

47
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square root of 4 is equal to x.

48
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I don't like this green color so much.

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And what's the square root of 4, the principal root of 4?

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It's 2.

51
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So that right there is 2.

52
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So this side becomes 2, this 2, times the square root of 2.

53
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And that is equal to x.

54
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Now let's verify that this is the solution to

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our original equation.

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So let's substitute this in, first to the left-hand side.

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So on the left-hand side, we have 5 times 2 square roots of

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2 squared minus 8.

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And then we're going to have to take the square root of

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that whole thing.

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So this is going to be equal to-- we're just focused on the

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left-hand side right now.

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This is equal to the square root of 5 times 2 squared,

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which is 4, times the square root of 2 squared, which is 2.

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And then minus 8.

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And this is 5 times 4 is 20 times 2 is 40.

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And then you have 40 minus 8 is 32.

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So this is equal to the square root of 32.

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Square root of 32 is the same thing as the square

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root of 16 times 2.

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The square root of 16 is 4.

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So this is the same thing as the square root of 16 times

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the square root of 2.

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Or 4 square roots of 2.

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So that's what the left hand simplifies to when we-- and

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remember, the original equation didn't have these

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squares here, so if you just look at the green part, the

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green part on the left-hand side just simplified to 4

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roots of 2.

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Let's see what 2x simplifies to.

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Our original right-hand side was just the 2x.

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That's parentheses with the square added later.

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So what's 2x?

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2 times 2 roots of 2.

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2 square root of 2.

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Well that's just 4 square root of 2.

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So when x is equal to 2 square roots of 2, the left-hand side

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equals 4 square roots of 2.

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And remember, the left-hand side looked like this when we

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started off.

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The left-hand side when we started off

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didn't have that there.

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I want to make that clear.

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So when you substitute this back into this left-hand side,

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you get 4 square roots of 2.

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When you substitute it back into the original right-hand

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side, you get 4 square roots of 2.

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So it's definitely the right-- I'm trying to write in black.

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It's definitely the right solution.

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