Lesson 4.15
Extraneous Solutions in Radicals
When you square both sides of an equation, you can create ghost solutions that satisfy the squared version but not the original. These are called extraneous solutions.
Introduction
Consider: . This has no solution (square roots are never negative). But if you square both sides, you get . That's extraneous — it was created by squaring. This lesson focuses entirely on detecting and rejecting these impostors.
Past Knowledge
Solving radical equations (4.13), domain of square roots.
Today's Goal
Identify extraneous solutions and determine when equations have no solution.
Future Success
Extraneous solutions also appear in logarithmic and rational equations later.
Key Concepts
Why Does This Happen?
Squaring is not reversible. Both and give 9 when squared.
So squaring produces a "solution" from the wrong sign branch.
Detection Strategy
Quick Check: Is the radical set equal to a negative?
→ immediate no solution
Full Check: Substitute back into the ORIGINAL equation
If both sides don't match → reject that solution
Worked Examples
Example 1: Obvious Extraneous
BasicSolve .
Stop immediately!
A principal square root is never negative. No need to square — there is no solution.
No solution (∅)
Example 2: Hidden Extraneous
IntermediateSolve .
Square both sides
Rearrange to standard form
or
Check both
x = 4:
x = −1:
only ( is extraneous)
Example 3: The Tricky Check
AdvancedSolve .
Square, rearrange, factor
Check both — don't skip this!
x = 3:
x = −2:
1 ≠ −1 → Extraneous!
Only
Common Pitfalls
Trusting Algebra Blindly
Both solutions from factoring can look "right" algebraically. The only way to know is to check in the original equation.
Checking in the Squared Version
You must check in the original equation (with the radical), not in the squared version. The squared version is precisely what created the extraneous solution.
Real-Life Applications
In physics, negative time values or negative distances often appear as extraneous solutions when modeling projectile motion or wave equations. Engineers must always check which algebraic solutions are physically meaningful.
Practice Quiz
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