#### 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 radical.
• If there are three 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 remaining radical.
• Raise both sides of the equation to a power equal to index of the isolated radical.
• You should now have a polynomial equation. Solve it.
• Rememer that you did not start out with a polynomial; therefore, there may be extraneous solutions. Therfore, 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. Therfore, the term is valid only if , the second term is valid only if , and the term is valid only if . The equation is valid if all the three terms are valid, therefore the domain is restricted to the common domain of the three terms or the set of real numbers .

Square both sides of the equation and simplify.

Isolate the term.

Square both sides of the equation and simplify.

Use the quadratic formula to solve for x.

The answers are .

Check the solution by substituting 1 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:

Since the left side of the original equation equals the right side of the original equation after we substituted our solution for x, then x=1 is a solution.

Check the solution by substituting -4.244444 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:

Since the left side of the original equation does not equal the right side of the original equation after we substituted -4.244444 for x, then the solution x= -4.244444 is not valid and not a solution after all.

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 is just on x-intercept at 1, then the only solution is x=1.

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.p> If you would like to go back to the equation table of contents, click on Contents.

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