EQUATIONS CONTAINING VARIABLES UNDER ONE OR MORE RADICALS

Note:


Problem 2.4d:

tex2html_wrap_inline135

Answer: tex2html_wrap_inline137 .

Solution: First make a note of the fact that you cannot take the square root of a negative number. The term tex2html_wrap_inline139 is valid only if tex2html_wrap_inline141 and the term tex2html_wrap_inline143 is valid if tex2html_wrap_inline145 . The restricted domain must satisfy both of these constraints. Therefore, the domain is the set of real numbers tex2html_wrap_inline147 . Multiply both sides of the equation by tex2html_wrap_inline149

eqnarray23





Since tex2html_wrap_inline151 is already isolated, square both sides of the equation.

eqnarray44





Isolate the tex2html_wrap_inline153 term.

displaymath131





Square both sides of the equation.

eqnarray54





Solve for x using the quadratic formula.

eqnarray63





There are two answers, one exact answer of 7 and one approximate answer of . 7.85207.




Check the solution 7 by substituting 7 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, we have verified that the solution is x=7.





Check the solution 7.85207 by substituting 7.85207 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, we have verified that the solutions is x=7.85207.





You can also check the answer by graphing the equation:

displaymath132

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 the x-intercepts are 7 and 7.85207, we have verified the solution.


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