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.

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 x-intercept 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.

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