5 min read
The Three Cases
x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-real.
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The Three Cases
What is it?
x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-real.
In simple words
- x2-6x+9=0 has D=0, repeated 3.
- sign of D
- D>0 two reals
- D=0 repeated
Important terms
The Three Cases: x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-real.
What you should learn
- Define and explain the three cases.
- Apply the three cases to a focused FBISE question.
- Check the answer for correct terms, evidence, working, and units.
Examples
- sign of D
- D>0 two reals
- D=0 repeated
CORE UNDERSTANDING
x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-real.
EXAM APPLICATION
State the correct term, connect it to the chapter rule or evidence, and use a clear example or calculation.
Recommended for Class 10
Full school-level explanation
x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-real.
Learn the idea, then use it:
WHAT — THE THREE CASES
x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-real.
HOW — Use the idea
Identify what is given, select this topic's rule or evidence, apply it one step at a time, and check the result.
CORE UNDERSTANDING
x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-real.
Remember
- sign of D
- D>0 two reals
- D=0 repeated
Examples
- sign of D
- D>0 two reals
- D=0 repeated
- THE THREE CASES: x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-real.
EXAM APPLICATION
State the correct term, connect it to the chapter rule or evidence, and use a clear example or calculation.
Remember
- correct term
- chapter link
- clear application
Examples
- Define first
- Show the rule or evidence
- Check units, labels, and conclusion
Remember
- sign of D
- D>0 two reals
- D=0 repeated
- keep the ideas separate
- sign of D
- D>0 two reals
- D=0 repeated
- correct term
Deeper understanding
Go beyond memorising The Three Cases. Connect the definition to reasons, relationships, exceptions, and exam wording.
Common mistakes
Exam focus
Core Idea
Key Points
Simple Check
How to Answer
Exam focus
Quick revision
- sign of D
- D>0 two reals
- D=0 repeated
- keep the ideas separate
- The Three Cases: x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-
- WHAT — THE THREE CASES: x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-
Do not merge the three cases with a neighbouring idea that uses a different definition, rule, or method.
For long answers on The Three Cases: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- sign of D
- D>0 two reals
- D=0 repeated
- keep the ideas separate
Important formulas
- sign of D
- D>0 two reals
- D=0 repeated
Common exam traps
- Do not merge the three cases with a neighbouring idea that uses a different definition, rule, or method.
- Do not invent a formula, fact, unit, or diagram label that the chapter does not support.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Do not merge the three cases with a neighbouring idea that uses a different definition, rule, or method.
- Mistake
Do not invent a formula, fact, unit, or diagram label that the chapter does not support.
- Mistake
Do not merge the three cases with a neighbouring idea that uses a different definition, rule, or method.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
sign of D
D>0 two reals
D=0 repeated
keep the ideas separate
x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-real.
D=0 repeated
- Definition
- x^2-6x+9=0 has D=0, repeated 3. x^2-3x+1=0 has D=5, two irrational reals. x^2+2x+5=0 has D=-16, non-real.
- Formula
- D=0 repeated
- Example
- sign of D
- Critical warning
- Do not merge the three cases with a neighbouring idea that uses a different definition, rule, or method.
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Exam Practice
Define
Q1. Define The Three Cases.
Explain
Q2. Explain: CORE UNDERSTANDING.
Example
Q3. Give one clear example of The Three Cases and say why it fits.
Mistake
Q4. State one common mistake students make about The Three Cases.
Exam note
Q5. Write a short exam note on The Three Cases: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.