Solution a14889a - Radical Equation.

Problem: Solve for real numbers x the equation
Solution:

This is a standard radical equation problem where an extraneous root may be introduced unless we pay careful attention to the domain for each step in the equation solving process.

Looking at this equation we make note of the domain restrictions imposed by the radicals:

Since both sides of the equation clearly are positive, we can square both sides without introducing extraneous solutions:

Here we again look at the situation and notice that the right side radical is not negative and we make note of this domain restriction:

With this additional domain restriction we can safely square both sides without introducing extraneous solutions:

The first root is clearly outside the domain for x which was determined above to be

and can be discarded. The answer therefore is:

Left and right sides with solution point.

Comment:

Had we not paid attention to the domain restrictions due to the square roots requiring non-negative contents on the inside and being non-negative on the outside, we might have included the extraneous root −5.8 in the answer. However, looking at the graph, this would have made no sense at all.

We might ask why the radicals have this non-negative requirements when for example

This does of course not apply to the square roots for which the non-negative argument works well. But should we treat cube roots and square roots differently when solving equations?

When it comes to generalizing the root concept we get into the area of exponential functions via the realization that:

Using fractional exponents eventually leads to this seriously questionable result:

This is one of the reasons why the convention has been generally adopted that, when working in the real numbers domain, any radicals are non-negative both inside and out.