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.

Problem 2.4d:

Answer: .

Solution: First make a note of the fact that you cannot take the square root of a negative number. 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 . Multiply both sides of the equation by

Since is already isolated, 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 answers, one exact answer of 7 and one approximate answer of . 7.85207.

Check the solution 7 by substituting 7 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 solution is x=7.

Check the solution 7.85207 by substituting 7.85207 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=7.85207.

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 7 and 7.85207, we have verified the solution.

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