A radical equation is an equation in which the variable appears inside a root.
The most common examples use square roots, but cube roots and higher-index roots also appear.
The main challenge is that raising both sides to a power can create extra candidates, so every result must be verified.
1. What is a radical equation?
Examples of radical equations include:
\[
\begin{aligned}
\sqrt{x+4}+\sqrt{x}&=4,\\
\sqrt{2x+3}+4&=7,\\
\sqrt[3]{x-2}&=3.
\end{aligned}
\]
A radical equation is solved by isolating a root, raising both sides to a power, solving the resulting equation,
and then checking every candidate in the original equation.
2. Domain restrictions
For real-number solutions, square roots and other even-index roots require nonnegative radicands:
\[
\begin{aligned}
\sqrt{A(x)}\text{ is real only if }A(x)\ge 0.
\end{aligned}
\]
More generally:
\[
\begin{aligned}
\sqrt[n]{A(x)}\text{ requires }A(x)\ge0
\quad\text{when }n\text{ is even.}
\end{aligned}
\]
Cube roots and other odd-index roots can accept negative radicands:
\[
\begin{aligned}
\sqrt[3]{-8}=-2.
\end{aligned}
\]
3. Single square-root equations
For an equation of the form:
\[
\begin{aligned}
\sqrt{A(x)}=B(x),
\end{aligned}
\]
square both sides:
\[
\begin{aligned}
A(x)=B(x)^2.
\end{aligned}
\]
Since the principal square root is never negative, the right side before squaring must also be nonnegative:
\[
\begin{aligned}
B(x)\ge0.
\end{aligned}
\]
4. Worked example
Solve:
\[
\begin{aligned}
\sqrt{2x+3}+4=7.
\end{aligned}
\]
First isolate the radical:
\[
\begin{aligned}
\sqrt{2x+3}=3.
\end{aligned}
\]
Square both sides:
\[
\begin{aligned}
2x+3&=3^2,\\
2x+3&=9.
\end{aligned}
\]
Solve:
\[
\begin{aligned}
2x&=6,\\
x&=3.
\end{aligned}
\]
Check in the original equation:
\[
\begin{aligned}
\sqrt{2(3)+3}+4
&=
\sqrt{9}+4\\
&=
3+4\\
&=
7.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{x=3}.
\end{aligned}
\]
5. Why extraneous solutions happen
Squaring both sides can hide sign information. For example:
\[
\begin{aligned}
\sqrt{x}=-2.
\end{aligned}
\]
If we square both sides:
\[
\begin{aligned}
x=4.
\end{aligned}
\]
But checking the original equation gives:
\[
\begin{aligned}
\sqrt{4}=2\ne -2.
\end{aligned}
\]
So \(x=4\) is an extraneous solution. The original equation has no real solution.
6. Equations with two radicals
Some equations contain two square roots:
\[
\begin{aligned}
\sqrt{x+4}+\sqrt{x}=4.
\end{aligned}
\]
A common strategy is:
- Move one radical to the other side.
- Square both sides.
- Isolate the remaining radical.
- Square again.
- Verify all candidates in the original equation.
Verification is especially important because two squaring steps can create extra candidates.
7. Cube roots and odd roots
Cube-root equations are often simpler because cubing both sides preserves signs:
\[
\begin{aligned}
\sqrt[3]{A(x)}=B(x)
\quad\Longrightarrow\quad
A(x)=B(x)^3.
\end{aligned}
\]
For example:
\[
\begin{aligned}
\sqrt[3]{x-2}=3.
\end{aligned}
\]
Cube both sides:
\[
\begin{aligned}
x-2&=27,\\
x&=29.
\end{aligned}
\]
8. Nested radical equations
A nested radical contains a root inside another root:
\[
\begin{aligned}
\sqrt{\sqrt{x}+1}=2.
\end{aligned}
\]
Square once:
\[
\begin{aligned}
\sqrt{x}+1=4.
\end{aligned}
\]
Isolate the remaining radical:
\[
\begin{aligned}
\sqrt{x}=3.
\end{aligned}
\]
Square again:
\[
\begin{aligned}
x=9.
\end{aligned}
\]
Then check in the original equation.
9. Graph interpretation
A radical equation:
\[
\begin{aligned}
\text{LHS}(x)=\text{RHS}(x)
\end{aligned}
\]
can be graphed as two curves:
\[
\begin{aligned}
y_1&=\text{LHS}(x),\\
y_2&=\text{RHS}(x).
\end{aligned}
\]
Real solutions occur where the two curves intersect. However, graphing is a visual check, not a replacement for exact algebra and substitution verification.
10. Formula summary
The table uses plain text formulas in the cells to avoid raw LaTeX issues in narrow layouts.
11. Common mistakes
- Squaring before isolating the radical.
- Forgetting the domain condition for square roots and even roots.
- Reporting every squared-equation solution without checking it in the original equation.
- Assuming cube roots require nonnegative radicands.
- Losing solutions because of arithmetic errors after squaring.
- Trusting a graph without doing substitution verification.
Key idea: isolate the radical, raise both sides to the correct power, solve the resulting equation, and always verify candidates.
