EQUATIONS CONTAINING VARIABLES UNDER ONE OR MORE RADICALS

Recall the following:


Solve for x in the following equation.

Problem 2.5c:

displaymath158

Answer: tex2html_wrap_inline164

Solution:
First make a note of the fact that you cannot take the square root of a negative number. Therefore, the tex2html_wrap_inline166 term is valid only if tex2html_wrap_inline168 , the term tex2html_wrap_inline170 is valid if tex2html_wrap_inline172 , and the term tex2html_wrap_inline174 is valid only if tex2html_wrap_inline176 . 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_inline178.




Square both sides of the equation and simplify.

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Isolate the tex2html_wrap_inline180 term and simplify.

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Square both sides of the equation and simplify.

eqnarray59




Use the quadratic formula to solve for x.

eqnarray71


The answers are tex2html_wrap_inline182 and tex2html_wrap_inline184 (rounded).




Check the solution by substituting 8.165254 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 8.165254 for x, then x=8.165254 is a solution.




Check the solution by substituting 4.890302 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 4.890302 for x, then x=4.890302 is a solution.



You can also check the answer by graphing the equation:

displaymath160

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 are the solutions. There are two x-intercepts, 4.890302 and 8.165254.


If you would like to review the solution to problem 2.5d, click on solution.

If you would like to go back to the problem page, click on problem.

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Author:Nancy Marcus

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