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Extraneous and Lost Solutions
Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original equation.
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Extraneous and Lost Solutions
What is it?
Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original equation.
In simple words
- Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original equation.
- Worked pattern: Never divide by sin x before separately testing sin x=0.
- Final check: domain, sign, notation, units, and reasonableness.
Important terms
Extraneous and lost solutions: Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original equation.
What you should learn
- State the rule for extraneous and lost solutions.
- Solve a extraneous and lost solutions problem step by step.
- Check the result, restrictions, notation, and units.
Recommended for Class 12
Full school-level explanation
Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original equation.
CONCEPT AND RULE
Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original equation.
METHOD
For extraneous and lost solutions, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
Never divide by sin x before separately testing sin x=0.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve extraneous and lost solutions
How it works
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Remember
- Extraneous and lost solutions
- rule
- worked example
- worked example
How the ideas connect
Worked example: Never divide by sin x before separately testing sin x=0.
Deeper understanding
Go beyond memorising Extraneous and Lost Solutions. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked example: Never divide by sin x before separately testing sin x=0.
Common mistakes
Exam focus
Core Idea
Rule / Formula
Method
Worked Example
Exam focus
Quick revision
- Extraneous and lost solutions
- rule
- worked example
- Extraneous and lost solutions: Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original
- CONCEPT AND RULE: Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original
- METHOD: For extraneous and lost solutions, identify the given information and required result. Select the ma
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
For long answers on Extraneous and Lost Solutions: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- Extraneous and lost solutions
- rule
- worked example
- Extraneous and lost solutions: Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original
Important formulas
- Extraneous and lost solutions
- rule
- worked example
Common exam traps
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- A matching-looking formula is not valid until its conditions and domain have been checked.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- Mistake
A matching-looking formula is not valid until its conditions and domain have been checked.
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Extraneous and lost solutions
rule
worked example
Extraneous and lost solutions: Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original
Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original equation.
- Definition
- Squaring can introduce roots; dividing by a trig factor can lose roots. Substitute into the original equation.
- Critical warning
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all d
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Exam Practice
Define
Q1. Define Extraneous and Lost Solutions.
Explain
Q2. Explain: CONCEPT AND RULE.
Example
Q3. Give one clear example of Extraneous and Lost Solutions and say why it fits.
Mistake
Q4. State one common mistake students make about Extraneous and Lost Solutions.
Exam note
Q5. Write a short exam note on Extraneous and Lost Solutions: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.