EQUATIONS CONTAINING VARIABLES UNDER ONE OR MORE RADICALS

Recall the following:


Solve for x in the following equation.

Problem 2.5d:

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Answer: x=10.

Solution

First make a note of the fact that you cannot take the square root of a negative number. Therefore,the tex2html_wrap_inline266 term is valid only if tex2html_wrap_inline268 , the term tex2html_wrap_inline270 is valid if tex2html_wrap_inline272 , and the term tex2html_wrap_inline274 is valid only if tex2html_wrap_inline276 . 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_inline278

Square both sides of the equation and simplify.

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Isolate the 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=10 and 0.547249 (rounded).




Check the solution by substituting 10 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 10 for x, then x=10 is a solution.




Check the solution by substituting 0.547249 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 0.547249 for x, then the solution x=0.547249 is a not valid and not a solution after all.




You can also check the answer by graphing the equation:

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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 10, then the only solution is x=10.


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Author:Nancy Marcus

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