Chapter 1.4, Problem 1E

To determine

To explain:

Extraneous solution.

Answer to Problem 1E

  $1$ is an extraneous solution.

Explanation of Solution

Extraneous solution are an apparent solutions that do not actually make the original equation true.

Consider the equations,

  $\sqrt{x}=-1$

For solving the given radical equation use the power rule, raise both sides of equation to the same integer power as root indexthen make use of principle.

  ${\left(\sqrt[n]{x}\right)}^n=x$

That eliminates the radical leaving its radicand.

Thus,

  $\begin{array}{l}\sqrt{x}=-1\\ {\left(\sqrt{x}\right)}^2={\left(-1\right)}^2\end{array}$

The root index is $2$ , so square both sides and solve for $x$ .

  $x=1$

For checking,

  $\begin{array}{l}{\sqrt{1}}^{?}=-1\\ 1=-1\end{array}$

So it is false.

So, 1is an extraneous solution.