EQUATIONS CONTAINING VARIABLES UNDER ONE OR MORE RADICALS
Note:
 In order to solve for x, you must isolate x.
 In order to isolate x, you must remove it from under the
radical.
 If there is just one radical in the equation, isolate the
radical.
 Then raise both sides of the equation to a power equal to
the index of the radical.
 With these types of equations, sometimes there are
extraneous solutions; therefore, you must check your answers.
 If the index of the radical is even, many times there will
be a restriction on the values of x.
Example 3:
First make a note of the fact that you cannot take the square root of
a negative number. Therefore, .
Subtract 5 from both sides of the equation so that the radical term is
isolated.
Square both sides of the equation:
Subtract 8x and 5 from both sides of the equation.
Solve using the quadratic formula.
Simplify.
Check the solution x=3.605958 by
substituting 3.605958 in the original equation. If after the
substitution, the left side of the original equation equals the right
side of the original equation, your answer is correct.

Left Side:
 Right Side:
(3.605958) = 10.817874
Check the solution
in the original equation. If after the substitution, the left side of
the original equation equals the right side of the original equation,
your answer is correct.  Left Side:
 Right Side: (0.616264) = 1.848792
Since the left
side of the original equation does not equal the right side of the
original equation when 0.616264 is substituted for x, then
0.616264 is not a solution. You can also check the answer by
graphing the equation:
The graph represents the
right side of the original equation minus the left side of the
original equation. Since the only xintercept is 3.605958,
x=3.615958 and therefore is the only solution.
If you would like to test yourself by working some problems similar
to this example, click on Problem.
If you would like to go back to the equation table of contents, click
on Contents.
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