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 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 4:

First make a note of the fact that you cannot take the square root of a negative number. Therefore, the tex2html_wrap_inline226 term is valid only if tex2html_wrap_inline228 the term tex2html_wrap_inline230 is valid if tex2html_wrap_inline232 , and the term is valid only if tex2html_wrap_inline236 The equation is valid if all three terms are valid, therefore the domain is restricted to the common domain of the three terms or the set of real numbers tex2html_wrap_inline238

Square both sides of the equation and simplify.

eqnarray41

Isolate the tex2html_wrap_inline240 term.

eqnarray95

Square both sides of the equation and simplify.

eqnarray102

Use the quadratic formula to solve for x.

eqnarray115

The answers are 15 and-0.078534.

Check the solution by substituting 15 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 15 for x, then x=15 is a solution.

Check the solution by substituting -0.078534 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:
is undefined because you cannot take the square root of a negative number and get a real answer.
Right Side:

Since the left side of the original equation does not equal the right side of the original equation after we substituted -0.078534 for x, then x=-0.078534 is not a real solution.

You can also check the answer by graphing the equation:

eqnarray150

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 one x-intercept at 15, then the only solution is x=15.

If you would like to test yourself by working some problems similar to this example, click on problem.

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

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Author:Nancy Marcus

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