Lesson 4.13
Solving Radical Equations
A radical equation has a variable inside a radical. The strategy: isolate the radical, then raise both sides to the power that eliminates it. Always check your answer!
Introduction
Equations like require a special approach. You can't just subtract or divide the radical away — you must square both sides (or cube, etc.) to undo the root. But squaring can introduce false solutions, so checking is essential.
Past Knowledge
nth roots (4.5), simplifying radicals (4.6), solving linear/quadratic equations.
Today's Goal
Solve equations with one radical by isolating and raising to a power.
Future Success
Lesson 4.15 focuses entirely on extraneous solutions — a critical skill from this lesson.
Key Concepts
The 4-Step Process
Isolate the radical on one side
Raise both sides to the nth power
Solve the resulting equation
Check your solution in the original
Power Rule
Raising an nth root to the nth power cancels the radical.
⚠️ Always check!
Squaring both sides can create solutions that don't work in the original equation (extraneous solutions).
Worked Examples
Example 1: Square Root Equation
BasicSolve .
Radical is already isolated. Square both sides.
Solve
Check
Example 2: Isolate First
IntermediateSolve .
Isolate the radical
Square both sides, then solve
Check
Example 3: Extraneous Solution
AdvancedSolve .
Radical is isolated. Square both sides.
Set equal to zero and factor
Two candidates: and
Check BOTH in the original
x = 4
x = −1
2 ≠ −2 → Extraneous!
Only
Common Pitfalls
Squaring Before Isolating
If you square without isolating first, you get a messy cross term. Isolate to first!
Skipping the Check
Squaring both sides can introduce extraneous solutions. Always substitute back into the original equation.
Real-Life Applications
The Pythagorean theorem is a radical equation in disguise. Finding missing sides or distances on a coordinate plane requires solving radical equations.
Practice Quiz
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