EQUATIONS CONTAINING VARIABLES UNDER ONE OR MORE RADICALS

Note:


Example 1:

tex2html_wrap_inline130

First make a note of the fact that you cannot take the square root of a negative number. Therfore, the tex2html_wrap_inline134 term is valid only if tex2html_wrap_inline136 , the second term tex2html_wrap_inline138 is valid only if tex2html_wrap_inline140 , and the term tex2html_wrap_inline142 is valid only if tex2html_wrap_inline144 . 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 tex2html_wrap_inline146 .




Square both sides of the equation and simplify.

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Isolate the tex2html_wrap_inline148 term.

displaymath96




Square both sides of the equation and simplify.

displaymath97

displaymath98

displaymath99

displaymath100

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Use the quadratic formula to solve for x.

displaymath102

displaymath103

displaymath104

displaymath105

displaymath106

displaymath107



The answers are tex2html_wrap_inline150.




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.

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.

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:

displaymath110

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.

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Author:Nancy Marcus

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