\begin{definition}{Radical Equation} An \textbf{equation} that \textbf{contains} a \textbf{variable} \textbf{inside} a \textbf{radical} \textbf{expression}. \end{definition}
\begin{note} \textbf{Radical equations} can have \textbf{multiple solutions} or \textbf{no solutions} at all. \end{note}
Solving Radical Equations
To solve a radical equation:
- \textbf{Isolate} the \textbf{radical} on \textbf{one} \textbf{side} of the equation.
- \textbf{Eliminate} the \textbf{radical} by \textbf{raising} both sides to the \textbf{power} of the \textbf{index} of the radical.
- \textbf{Solve} the \textbf{resulting} \textbf{equation}.
- \textbf{Check} for \textbf{extraneous} \textbf{solutions} by \textbf{substituting} back into the \textbf{original} \textbf{equation}.
\begin{warning} \textbf{Extraneous solutions} are \textbf{solutions} that \textbf{arise} from the \textbf{process} of \textbf{solving} the \textbf{equation} but \textbf{do not} \textbf{satisfy} the \textbf{original} \textbf{equation}. \end{warning}
\begin{example} Solve the equation $\sqrt{x + 3} = x - 1$.
- The radical is already isolated.
- Square both sides: $(\sqrt{x + 3})^2 = (x - 1)^2$.
- Simplify: $x + 3 = x^2 - 2x + 1$.
- Rearrange: $x^2 - 3x - 2 = 0$.
- Factor: $(x - 2)(x - 1) = 0$.
- Solutions: $x = 2$ or $x = 1$.
- Check solutions:
- For $x = 2$: $\sqrt{2 + 3} = 2 - 1 \Rightarrow 5 = 1$ (False)
- For $x = 1$: $\sqrt{1 + 3} = 1 - 1 \Rightarrow 2 = 0$ (False)
No solutions. \end{example}
\begin{self_review} Solve the equation $\sqrt{2x - 1} = x - 3$ and check for extraneous solutions. \end{self_review}
\begin{tok} How can we be sure that the solutions we find are correct? What does it mean for a solution to be "extraneous"? \end{tok}
