Solve rational and radical equations without extraneous roots
Record excluded denominators and root domains · Solve after clearing fractions or squaring · Test every candidate in the original equation
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Linear and quadratic equations
- Factorising and square roots
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Record excluded denominators and root domains
- Solve after clearing fractions or squaring
- Test every candidate in the original equation
Conditions come before transformations
A rational equation contains variable denominators. List the values that make them zero before multiplying through. Multiplying by a common denominator is reversible only where that denominator is non-zero.
For √g(x) = h(x), a real solution needs g(x) ≥ 0 and h(x) ≥ 0 because the principal square root cannot be negative. Squaring produces g(x) = h(x)², but can introduce candidates with the wrong sign.
Return to the original
Solve the transformed polynomial equation, then substitute each candidate in the original. A forbidden denominator or unequal original sides rejects the candidate. Do not reject a valid zero numerator when the denominator is non-zero.
If all candidates fail, say there is no real solution; an algebraic candidate is not automatically a solution.
Sources for this lesson
Pause and explain
Is every solution of a squared equation a solution of the original?
Worked example
Solve √(x + 6) = x.
Show the worked solution
- The right side must be non-negative, so x ≥ 0.
- Squaring gives x + 6 = x², or (x − 3)(x + 2) = 0.
- Candidates are 3 and −2. √9 = 3 works; √4 = 2 does not equal −2.
Answer x = 3 only; −2 is extraneous.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Solve 2/(x − 1) = 1, stating exclusions.
Give me a hint
First identify when the denominator is zero.
Compare my reasoning
- x ≠ 1. Multiply by x − 1 to get 2 = x − 1.
- x = 3 gives 2/(3 − 1) = 1.
x = 3, with x = 1 excluded.
Look for these in your work
- Record the restriction
- Substitute back
Solve √(2x + 3) = x.
Give me a hint
Require x ≥ 0 before squaring.
Compare my reasoning
- Squaring gives x² − 2x − 3 = (x − 3)(x + 1) = 0.
- 3 gives √9 = 3; −1 gives √1 = 1 ≠ −1.
x = 3 only; reject −1.
Look for these in your work
- Factor accurately
- Reject the sign failure
Solve (x² − 4)/(x − 2) = 4. Does cancelling justify x = 2?
Give me a hint
The original denominator remains part of the problem.
Compare my reasoning
- The domain excludes x = 2.
- For x ≠ 2, the fraction equals x + 2, so x + 2 = 4 gives candidate 2.
- That candidate is excluded; no other candidate exists.
No real solution. Cancellation does not permit x = 2.
Look for these in your work
- Retain the original exclusion
- Report an empty solution set clearly
Common mistakes
- Squaring loses sign information.
- Cancelling a factor does not restore a value excluded by the original denominator.
What can you explain now?
Is every solution of a squared equation a solution of the original?
Compare with the explanation
No; it must pass the original equation and domain checks.
Opposite signs can have equal squares.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Is every solution of a squared equation a solution of the original?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.