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 5:

Since the index of the radical is even, the quantity under the

radical cannot be negative and .

Subtract 18 from both sides of the equation so that the radical

term is isolated.

At this point you observe that the positive left side of the

equation cannot equal the negative number on the right side of the

equation, you are through. There is no solution. If you did not make

this observation, let's continue.

Divide both sides by 3 and then square both sides of the equation.

(2*x*+9)=4

2*x*=-5

The answer is

Check the solution by substituting 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.

Left side:

3*2+18

6+18

=24

Right side: Since the left side of the equation does not equal the right side of

the original equation when you substitute your answer, your answer is

not the solution. There is no solution.

You can also check the answer by graphing

(the left side of the original equation minus the right side of the

original equation). The solution will be the x-intercept. There is no x-

intercept, and hence there is no solution.

If you would like to test yourself by working some problems similar to

this example, click on problem Links to S2211

If you would like to go back to the equation table of contents, click

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