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.

Problem 2.3b:

Answer: (rounded)

Solution:

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

Subtract 9x from both sides of the equation so that the radical term is

isolated.

Square both sides of the equation:

Subtract 3x and 2 from both sides of the equation.

Solve using the quadratic formula.

Simplify.

Check the solution by substituting 1.386897 for x in the original equation. If after the substitution, the left side of the original equation equals the right side of the original equation, 1.386897 is a solution.

• Left Side:

• Right Side: 10

Since the left side of the original equation does not equal the right side of the original equation after 1.386897 was substituted for x, then x=1.386897 is not a solution.

Check the solution x= 0.872362 by substituting 0.872362 in the original equation. If after the substitution, the left side of the original equation equals the right side of the original equation, 0.872362 is a solution.

• Left Side:

• Right Side: 10

Since the left side of the original equation equals the right side of the original equation after 0.872362 was substituted for x, then x=0.872362 is 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. There is only one x-intercept, at 0.872362, therefore 0.872362 is a solution to the equation.

If you would like to review the answer and solution to problem 2.3c, click on Solution.

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