MathematicsChapter 2: Quadratic Equations
The Three Cases
x^2-6x+9=0 has D=0, repeated 3.
Learning objectives
- 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.
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=-16
Where / symbols
- 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.
Relationship
- D=0 repeated
Overview
Learn the idea, then use it:
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.
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.
Examples Define first · Show the rule or evidence · Check units, labels, and conclusion
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.
In simple words
Do not merge the three cases with a neighbouring idea that uses a different definition, rule, or method.
Key points
- sign of D
- D>0 two reals
- D=0 repeated
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.The Three Cases
Sticky Notes · Quick Revision
Core Idea
- x2-6x+9=0 has D=0, repeated 3.
Key Points
- sign of D
- D>0 two reals
- D=0 repeated
Simple Check
- Explain the three cases in your own words and connect it to one chapter example.
How to Answer
- Define the idea, show the main rule or evidence, apply it, and check the final statement.
Common Mistake
- Do not merge the three cases with a neighbouring idea that uses a different definition, rule, or method.
FBISE Exam Tip
- Use the exact term, show working where needed, and include units or labelled evidence.
60-second revision
- x2-6x+9=0 has D=0, repeated 3.
- sign of D
- D>0 two reals
- D=0 repeated
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