Radical equations
Solving Radical Equations 3 Solving Radical Equations 3
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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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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