#### EQUATIONS CONTAINING VARIABLES UNDER ONE OR MORE RADICALS

Note:

- In order to solve for x, you must isolate x.
- In order to isolate x, you must remove it from under the radial.
- If there are two radicals in the equation,isolate one of the radicals.
- Then raise both sides of the equation to a power equal to the index
of the isolated radical.
- Isolate the the remaining radical.
- Raise both sides of the equation to a power equal to the index of the
isolated radical.
- You should now have a polynomial equation. Solve it.
- Remember that you did not start out with a polynomial; therefore,
there may be extraneous solutions. Therefore, you must check your answers.

Example 1:

First make a note of the fact that you cannot take the square root of a
negative number. Therefore,the term is valid only if
and the second term is valid if

Isolate the term.

Square both sides of the equation.

Isolate the term.

Square both sides of the equation.

Check the solution by substituting 9 in the original equation for x. If
the left side of the equation equals the right side of the equation after
the substitution, you have found the correct answer.

- Left side:
- Right Side:1

Since the left side of the original equation does not equal the right side
of the original equation after we substituted our solution for x, then there
is no solution.

You can also check the answer by graphing the equation:

.The graph represents the right side of the original equation minus the left
side of the original equation.. The x-intercept(s) of this graph is(are) the
solution(s). Since there are no x-intercepts, there are no solutions.

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If you would like to review another example, click on Example
**
If you would like to test yourself by working some
problems similar to this example, click on Problem.

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[Algebra]
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[Geometry]
[Differential Equations]
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[Complex Variables]
[Matrix Algebra]

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**Author:Nancy Marcus**

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