#### 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 3:

First make a note of the fact that you cannot take the square root of a
negative number. Therefore,the first term is valid only if and the second term is valid if . Therefore, the domain
is restricted to the set of real numbers x, .

Isolate the term

Square both sides of the equation.

Isolate the term.

Square both sides of the equation.

Solve for x using the quadratic formula.

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

Check the solution 4 by substituting 4 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: 6

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*=4.

Check the solution -1.387755 by substituting -1.387755 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:6

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*=-1.387755.

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-intercepts are 4 and -1.387755, we have verified
the solutions.

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

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[Geometry]
[Differential Equations]
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[Complex Variables]
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