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 5:

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

Isolate the term by adding 3 to both sides of the equation.

Square both sides of the equation.

Isolate the term.

Square both sides of the equation.

Solve for x using the quadratic formula.

There are two approximate answers, 8163.162 and 18.8385.

Check the solution 8163.162 by substituting 8163.162 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=8163.162.

Check the solution 18.8385 by substituting 18.8385 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 18.8385 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 8163.162, we have verified the solution.

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

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