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 1:

First make a note of the fact that you cannot take the square root of a negative number. Therefore, .

Add 3x to both sides of the equation so that the radical term is isolated.

Square both sides of the equation:

Subtract x and 1 from both sides of the equation.

Factor the right side of the equation.

Solve for x.

Check the solution by substituting 0 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: 1

Since the left side of the original equation equals the right side of the original equation when 0 was substituted for x, this means that x=0 is a valid solution.

Check the solution by substituting 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: 1

Since the left side of the original equation did not equal the right side of the original equation when was substituted for x, this means that is not a valid solution.

The only solution is x=0.

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). You can see that the graph has an x-intercept of 0.

If you would like to work another example, click on Example.

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

If you would like to go back to the equation table of contents, click on Contents.

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