Solving Radical Equations
Learn how to solve equations containing square roots by isolating the radical and squaring both sides.
Definition
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Worked Examples
Solve:
The radical is already isolated
→ Ready to square
Square both sides
→
Check the solution
✓ → Valid solution
Answer:
Common Mistakes
Forgetting to check for extraneous solutions
Why it's wrong: Squaring both sides can introduce false solutions. For example, in , squaring gives , but .
Correct: Always substitute your answer back into the ORIGINAL equation to verify.
Squaring before isolating the radical
Why it's wrong: If the radical is not alone, squaring creates a more complex equation. For , squaring gives , which is harder to solve.
Correct: First isolate: , then square: .
Only squaring the radical, not the entire side
Why it's wrong: Both complete sides must be squared.
Correct: Square the entire expression:
Assuming square root can be negative
Why it's wrong: The principal square root is always non-negative. , not .
Correct: If where , there is no solution.
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Practice Problems
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Why It Matters
- Physics: The period of a pendulum is , where is the length
- Distance formula: Finding coordinates when
- Engineering: Stress calculations, wave equations, and electrical circuits
- Finance: Time value of money formulas involving roots
Real World Applications
Pendulum Period
The period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{L}{g}}$ where $L$ is the length and $g = 9.8$ m/s2 is gravity.
Example:
If a clock pendulum needs a period of 2 seconds, what length should it be? Solve
A grandfather clock needs a pendulum with a period of exactly 1 second.
What length pendulum is needed? Use
Step 1: Write the mathematical expression
Start with
Free Fall Distance
The distance an object falls is related to time by $d = \frac{1}{2}gt^2$, or equivalently $t = \sqrt{\frac{2d}{g}}$.
Example:
How long does it take for a ball to fall 45 meters? Solve
A rock is dropped from a cliff and takes 3 seconds to hit the ground.
How high is the cliff? Use
Step 1: Write the mathematical expression
Solve for
Key Takeaways
- 1A radical equation contains a variable under a radical sign (like )
- 2To solve: isolate the radical, square both sides, solve, then CHECK
- 3Squaring can create extraneous solutions that don't work in the original equation
- 4A square root equals a negative number has no real solution
- 5Always verify solutions by substituting back into the original equation
Frequently Asked Questions
Glossary
- Radical equation
- An equation where the variable appears under a radical sign
- Extraneous solution
- A value that satisfies the transformed equation but not the original equation
- Principal square root
- The non-negative square root of a number, denoted by
- Isolate
- To get a term alone on one side of an equation
Formula Card
Basic Method
Square both sides to eliminate the radical
Check Condition
The right side must be non-negative
Domain Restriction
Expression under the radical must be non-negative