Solving Radical Equations 3

U07_L3_T2_we3 Solving Radical Equations 3

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.