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.

Problem 2.1e:

Let's make a few observation right away. If there is no +or- sign in from

of the radical symbol, the sign is assumed to be positive. If you at first

observe that it is impossible for a positive number to equal a negative

number, you are through. There is no solution.

If you jumped right into the middle of the problem without checking, you

will also learn that there is no solution when you check the answer.

It will just take you a little work to reach the same conclusion that

there is no solution. If you do not check your answer, you are

gambling.

Raise both sides of the equation to the power 2.

Add 11 to both sides of the equation

The answer is x=5

Check the solution by substituting 5 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:

Right Side: -3

Conclusions:Since the left side of the equation does not equal

the right side of the equation after you substitute what you think

the solution is, the solution is wrong. Since the only solution you

found was 5 and 5 did not work, the conclusion is that there are no

solutions.

You can also check your answer with a graphing calculator. Graph the

following equation and check where it crosses the x-axis:

The above equation was formed by subtracting the right side of the

equation from the left side of the equation. You will note that the

graph never crosses the x-axis, which means there is not solution

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