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 radial.
• If there are two radicals in the equation,isolate one of the radicals.
• Then raise both sides of the equation to a power equal to the index of the isolated radical.
• Isolate the remaining radical.
• Raise both sides of the equation to a power equal to the index of the isolated radical.
• You should now have a polynomial equation. Solve it.
• Remember that you did not start out with a polynomial; therefore, there may be extraneous solutions. Therefore, you must check your answers.

Example 4:

First make a note of the fact that you cannot take the square root of a negative number. Therefore,the term is valid only if and the term is valid if . The restricted domain must satisfy both of these constraints. Therefore, the domain is the set of real numbers

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

Isolate the term.

Square both sides of the equation.

Simplify the equation by dividing both sides by 3.

Solve for x using the quadratic formula.

There is one exact answer of 4 and one approximate answer of -1.387755.

Check the solution -8 by substituting -8 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:

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=-8.

Check the solution -37.333 by substituting -37.333 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:

Since the left side of the original equation does not equal the right side of the original equation after we substituted our solution for x, the solution -37.333 is not a valid 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. The x-intercept(s) of this graph is(are) the solution(s). Since the x-intercept is -8, we have verified the solution.

If you would like to review another example, click on Example

If you would like to test yourself by working some problems similar to this example, click on Problem.

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