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 2.3d:

Answer: No Solution.

Solution:

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

Subtract 8 from both sides of the equation so that the radical term is isolated.

Square both sides of the equation:

Subtract x from and add 7 to both sides of the equation.

Solve using the quadratic formula.

(rounded)

The answers are x=4.724533, x=3.386578(rounded)

Check the solution x=4.724533 by substituting 4.724533 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,4.724533 is a solution.

• Left Side:

  Since you cannot take the square root of a negative number, x=4.724533 is not a valid solution.

• Right Side: (4.724533) = 9.449066

Check the solution x=3.386578 by substituting 3.386578 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, 3.386578 is a solution.

• Left Side:

  Since you cannot take the square root of a negative number, x=3.386578 is not a valid solution.

• Right Side: (3.386578) = 6.773156

Since the left side of the original equation does not equal the right side of the original equation after 3.386578 was substituted for x, then x=3.386578 is not 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. You can see that there are no x-intercepts. This means that there are no solutions.

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

If you would like to go back to the problem page, click on Problem.

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

[Algebra] [Trigonometry] [Geometry] [Differential Equations] [Calculus] [Complex Variables] [Matrix Algebra]

S.O.S. MATHemathics home page