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- Let's do a few more examples of solving radical equations.
- Let's say I have the square root of x plus 2 minus 2 is
- equal to 0.
- So just like I said in the last video, we always want to
- isolate the radical.
- And the best way I can think of doing it over here, is
- adding 2 to both sides of this equation.
- So if you add 2 to both sides of this equation, we are left
- with the square root of x plus 2-- on the left-hand side this
- cancels out --is equal to 2.
- And once you isolate one of the radicals, then you're
- ready to essentially, square both sides of this.
- If we're dealing with a square root, we square
- both sides of this.
- Let's do that.
- So let's square both sides of that equation.
- And then we get-- square root of x plus 2 squared is just x
- plus 2 is equal to 2 squared.
- Is equal to 4.
- Subtract 2 from both sides of this equation.
- We're left with x-- these 2's cancel out --is equal to 2.
- Now let's validate that this is truly the case.
- That it works when we're taking the principal root, not
- both square roots or the negative square root.
- Let's see, when we take the principal root we get the
- square root of 2-- that's our answer --plus this 2, minus 2,
- should be equal to 0.
- This is 4.
- The principal root of 4 is 2.
- 2 minus 2 does indeed equal 0.
- So this is a valid solution.
- It's not an extraneous solution.
- It works with the principal root.
- Let's do one that doesn't involve a square root.
- Let's do one with a higher power root.
- Let's say we have the fourth root of x squared minus 9 is
- equal to 2.
- So you say, gee, Sal, what do I do here?
- Fourth root?
- That's crazy.
- How do we deal with the fourth root?
- And it's the same process.
- When you had a square root we squared it.
- When you have the fourth root, we take it
- to the fourth power.
- And just as a bit of a review, this could be completely
- written like this.
- Another way to write it would be x squared minus 9 to the
- 1/4 power is equal to 2.
- These are equivalent statements, just different
- ways of writing the fourth root.
- Now, if we raise both sides to the fourth power, what happens
- to this exponent right here?
- If we raise something to an exponent, and then raise that
- to another exponent, we multiply the exponents.
- So 1/4 times 4 is 1.
- So this just becomes x squared minus 9 to the first power.
- Or we could just write x squared minus 9.
- And that's going to be equal to 2 to the fourth power.
- 2 to the fourth power is, what?
- 2 times 2 is 4.
- Times 2 is 8.
- Times 2 is 16.
- There's two ways we can do it actually at this point.
- We can add 9 to both sides of this equation.
- I'll do it this way in yellow.
- So if you add 9 you get x squared is equal to 25.
- Take the square root, not just the principal square root,
- take the positive and negative square root of both sides of
- this equation and you get x is equal to plus or minus 5.
- That's one way.
- The other way-- sometimes it's confusing, people forget to
- take the positive and negative square roots --is to do it the
- way we've been doing all along through solving traditional
- quadratics, setting them equal to 0.
- So what you do is you subtract 16 from both
- sides of this equation.
- If you do that you get x squared minus 25, if we
- subtract 16 from both sides, is equal to 0.
- So these two are equivalent.
- Essentially, we're just moving the 25 onto
- the left-hand side.
- We're just subtracting 25 from both sides
- of this to get that.
- But what this does is it puts us in the form of a difference
- of squares.
- We know we can rewrite this as x plus 5 times x minus 5.
- We went over that several videos ago.
- And that will be equal to 0.
- And if the product of two numbers is equal to 0, that
- means that either one or both of them is equal to 0.
- So x plus 5 is equal to 0 or x minus 5 is equal to 0.
- And we get x is equal to negative 5 or x is equal to
- positive 5, which is what we got right there.
- Now, the one thing we want to make sure once again, is that
- neither of these are extraneous solutions.
- Because when you're taking a fourth root or when you take
- something to the fourth power, negative
- numbers also disappear.
- You can kind of view it as a principal fourth root that
- we're taking here.
- A negative version of this could have also been valid.
- So let's make sure that they both work.
- Well, whether we put negative 5 or positive 5 here, this
- right here is going to be 25.
- So you get the fourth root of 25 minus 9, which is the
- fourth root of 16.
- So this is the fourth root of 16, which is equal to 16 to
- the 1/4 power.
- Which is indeed, 2.
- So both of these work.
- When you square either of these, you do indeed, get 25.