1. Solving Radical Equations

2. Extraneous Solutions
Power Rule (text only talks about squaring, but applies to other powers, as well).
If both sides of an equation are raised to the same power, solutions of the new equation contain all the solutions of the original equation, but might also contain additional solutions.
A proposed solution of the new equation that is NOT a solution of the original equation is an extraneous solution.

3. Solving Radical Equations
Steps for Solving Radical Equations
Isolate one radical on one side of equal sign.
Raise each side of the equation to a power equal to the index of the isolated radical, and simplify. (With square roots, the index is 2, so square both sides.)
If equation still contains a radical, repeat steps 1 and 2. If not, solve equation.
Check proposed solutions in the original equation.

4. that contains a radical.
A radical equation
is an equation
that contains a radical.

5. mathematical definitions
Did you ever notice
how much sense
mathematical definitions
usually make?

6. in solving most equations.
The goal in solving
radical equations
is the same as the goal
in solving most equations.

7. We need to isolate
the variable.

8. But there is only one way
to move the variable
out from under the
square root sign.

9. We need to square the
radical expression.

10. And, because it is an
equation,
what we do to one side,

11. we have to do to the other.
And, because it is an
equation,
what we do to one side,
we have to do to the other.

12. And now, we need
to simplify:

13. (Even if n is an expression)
Remember,
no matter what
n is.
(Even if n is an expression)

14. So we have:

15. So we have:
So we have:

16. Solving Radical Equations
Solve the following radical equation.
Substitute into the original equation.
true
So the solution is x = 24.

17. Solving Radical Equations
Solve the following radical equation.
Substitute into the original equation.
Does NOT check, since the left side of the equation is asking for the principal square root.
So the solution is .