Basic Technical Mathematics, 11th Edition

14.4 Equations with Radicals

• Solving by Squaring Both Sides • Isolating a Radical • Solving a Nested Radical Equation

Equations containing radicals are normally solved by squaring both sides of the equation if the radical represents a square root or by a similar operation for the other roots. However, when we do this, we often introduce extraneous roots.

CAUTION

Thus, it is very important that all solutions be checked in the original equation.

EXAMPLE 1 Solve by squaring both sides

Solve the equation $\sqrt{x\hspace{0.17em}-\hspace{0.17em}4}\hspace{0.17em}=\hspace{0.17em}2.$

By squaring both sides of the equation, we have

$$\begin{array}{rcl}\hfill {(\sqrt{x\hspace{0.17em}-\hspace{0.17em}4})}^2& \hspace{0.17em}=\hspace{0.17em}& 2^2\\ \hfill x\hspace{0.17em}-\hspace{0.17em}4& \hspace{0.17em}=\hspace{0.17em}& 4\\ \hfill x& \hspace{0.17em}=\hspace{0.17em}& 8\end{array}$$

This solution checks when put into the original equation.

EXAMPLE 2 Solve by squaring both sides

Solve the equation $2\sqrt{3x\hspace{0.17em}-\hspace{0.17em}1}\hspace{0.17em}=\hspace{0.17em}3x\hspace{0.17em}.\hspace{0.17em}$

Squaring both sides of ...