Radical Equation Examples

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Radical Equation Examples

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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.