EQUATIONS CONTAINING VARIABLES UNDER ONE OR MORE RADICALS

Recall the following:


Solve for x in the following equation.

Problem 2.5a:

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Answer: x=1.

Solution
First make a note of the fact that you cannot take the square root of a negative number. Therefore,the tex2html_wrap_inline140 term is valid only if tex2html_wrap_inline142 , the term tex2html_wrap_inline144 is valid if tex2html_wrap_inline146 , and the term tex2html_wrap_inline148 is valid only if tex2html_wrap_inline150 . 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_inline152.

Square both sides of the equation and simplify.

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Isolate the tex2html_wrap_inline154 term and simplify.

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Square both sides of the equation and simplify.

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Use the quadratic formula to solve for x.

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The answers are x=1, -0.017995.




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 1 for x, then x=1 is a solution.




Check the solution by substituting -0.017995 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 -0.017995 for x, then the solution x=-0.017995 is a not valid and not a solution after all.




You can also check the answer by graphing the equation:

displaymath136

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 1, then the only solution is x=1.


If you would like to review the solution to problem 2.5b, 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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