Radical Equations
By the end of this lesson, you’ll be able to:
- Solve equations involving square roots and higher roots.
- Identify and reject extraneous solutions.
- Use squaring as a tool to eliminate radicals safely.
Key Ideas
Solving radical equations generally follows the same pattern:
- Isolate the radical.
- Square both sides to remove the radical.
- Solve the resulting equation.
- Check solutions to eliminate extraneous results.
Example structure:
\[ \sqrt{ax + b} = c \]
After squaring:
\[ ax + b = c^2 \]
Checking is essential because squaring can introduce false solutions.
Common Problem Types
1. Basic Radical Equations
A single radical equals a number; isolate and square.
2. Radical Equals an Expression
Squaring may create linear or quadratic equations.
3. Equations Producing Extraneous Solutions
Even correct algebra may produce answers that fail the original equation.
4. “No Solution” Situations
If isolating leads to \(\sqrt{\text{something}} = \text{negative number}\) → no solution.
Strategies
- Always isolate the radical before squaring.
- Square carefully: apply \(\left(\dots\right)^2\) to each side fully.
- Expect quadratics—solve using factoring or the quadratic formula.
- Substitute each solution back into the original equation.
- Reject any solution that makes the radical negative or the equation false.
Worked Examples
Example 1 — Basic Radical Equation
Solve:
\[
\sqrt{x + 3} = 5
\]
Solution:
\[
\begin{split}
x + 3 &= 25 \\
x &= 22
\end{split}
\]
Check: \(\sqrt{22 + 3} = 5\) ✔
Example 2 — Extraneous Solution Appears
Solve:
\[
\sqrt{2x - 1} = x - 3
\]
Solution:
Square both sides:
\[ 2x - 1 = (x - 3)^2 \]Expand the right side:
\[ 2x - 1 = x^2 - 6x + 9 \]Rearrange:
\[ x^2 - 8x + 10 = 0 \]
Solve the quadratic:
\[
x = 5, \quad x = 2
\]
Check both:
- \(x = 5\) works
- \(x = 2\) gives \(\sqrt{3} \neq -1\) → extraneous ❌
- Failing to check solutions for extraneous results.
- Squaring before isolating the radical, creating unnecessary complexity.
- Overlooking that \(\sqrt{\text{expression}}\) cannot equal a negative number.
Practice Problems
- Solve: \(\sqrt{x - 4} = 6\)
- Solve: \(\sqrt{3x + 1} = x\)
- Solve: \(\sqrt{5x} = x - 1\)
- Solve: \(\sqrt{x + 9} = x + 1\)
- Solve: \(\sqrt{2x + 3} = 4\)
1.
\[
\begin{split}
x - 4 &= 36 \\
x &= 40
\end{split}
\]
2.
\[
\begin{split}
3x + 1 &= x^2 \\
x^2 - 3x - 1 &= 0 \\
x &= 3 \quad (\text{only solution that checks})
\end{split}
\]
3.
\[
\begin{split}
5x &= (x - 1)^2 \\
5x &= x^2 - 2x + 1 \\
x^2 - 7x + 1 &= 0 \\
x &= 5 \text{ (checks)}
\end{split}
\]
4.
\[
\begin{split}
x + 9 &= (x + 1)^2 \\
x + 9 &= x^2 + 2x + 1 \\
x^2 + x - 8 &= 0 \\
x &= 3 \text{ (checks)}
\end{split}
\]
5.
\[
\begin{split}
2x + 3 &= 16 \\
x &= 6
\end{split}
\]
Summary
- Isolate the radical before squaring.
- Squaring both sides eliminates the radical and may create quadratics.
- Always check solutions to remove extraneous ones.
- A radical cannot equal a negative number—this indicates no solution.
- Careful checking ensures only valid solutions remain.
- Isolate first, square second—never reverse the order.
- After squaring, expect additional steps (simplifying, factoring, solving).
- Plug solutions back into the original equation, not the squared one.
- Any solution that makes the radical negative must be rejected.