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Special Factors
Difference of squares: x^2 - 9 = (x-3)(x+3), so x = +/- 3. Perfect square: x^2 + 6x + 9 = (x+3)^2, a repeated root x=-3. These are still quadratics; they just factor quickly.
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Special Factors
What is it?
Difference of squares: x^2 - 9 = (x-3)(x+3), so x = +/- 3. Perfect square: x^2 + 6x + 9 = (x+3)^2, a repeated root x=-3. These are still quadratics; they just factor quickly.
In simple words
- Difference of squares: x2 - 9 = (x-3)(x+3), so x = +/- 3.
- DIFFERENCE OF SQUARES: x2 - 9 = 0 -> (x-3)(x+3)=0 -> x=3 or x=-3.
- PERFECT SQUARE: x^2 + 6x + 9 = (x+3)^2 = 0 gives x=-3 twice (repeated / equal roots).
- Worked-method reminder: x = -3 (repeated)
Important terms
SPECIAL FACTORS: Difference of squares: x^2 - 9 = (x-3)(x+3), so x = +/- 3. Perfect square: x^2 + 6x + 9 = (x+3)^2, a repeated root x=-3. These are still quadratics; they just factor quickly.
DIFFERENCE OF SQUARES: x^2 - 9 = 0 -> (x-3)(x+3)=0 -> x=3 or x=-3. Same idea for 4x^2 - 25 after you see (2x-5)(2x+5).
PERFECT SQUARE: x^2 + 6x + 9 = (x+3)^2 = 0 gives x=-3 twice (repeated / equal roots). Discriminant will be 0 on a later page, which matches.
What you should learn
- Explain special factors using the correct definitions, conditions, and notation.
- Apply x^2 - k^2, (x+k)^2 correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- x2-9=0 -> +/-3
- x2-16=0 -> +/-4
- Not (x-3)^2
DIFFERENCE OF SQUARES
x^2 - 9 = 0 -> (x-3)(x+3)=0 -> x=3 or x=-3. Same idea for 4x^2 - 25 after you see (2x-5)(2x+5).
PERFECT SQUARE
x^2 + 6x + 9 = (x+3)^2 = 0 gives x=-3 twice (repeated / equal roots). Discriminant will be 0 on a later page, which matches.
Recommended for Class 10
Full school-level explanation
Difference of squares: x^2 - 9 = (x-3)(x+3), so x = +/- 3. Perfect square: x^2 + 6x + 9 = (x+3)^2, a repeated root x=-3. These are still quadratics; they just factor quickly.
Two patterns to spot:
WHAT — SPECIAL FACTORS
Difference of squares: x^2 - 9 = (x-3)(x+3), so x = +/- 3. Perfect square: x^2 + 6x + 9 = (x+3)^2, a repeated root x=-3. These are still quadratics; they just factor quickly. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Special Factors matters because quadratic equations model areas, motion, and many algebraic relationships. Knowing more than one solution method lets you choose the quickest reliable method and check the nature of the roots.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through DIFFERENCE OF SQUARES, then PERFECT SQUARE. Worked: solve x^2 + 6x + 9 = 0. x^2 + 6x + 9 = 0 (x + 3)(x + 3) = 0 (x + 3)^2 = 0 x + 3 = 0 x = -3 (repeated) Finish by checking signs, restrictions, units, and whether the answer matches the question.
DIFFERENCE OF SQUARES
x^2 - 9 = 0 -> (x-3)(x+3)=0 -> x=3 or x=-3. Same idea for 4x^2 - 25 after you see (2x-5)(2x+5).
Remember
- +/- roots
Examples
- x2-9=0 -> +/-3
- x2-16=0 -> +/-4
- Not (x-3)^2
PERFECT SQUARE
x^2 + 6x + 9 = (x+3)^2 = 0 gives x=-3 twice (repeated / equal roots). Discriminant will be 0 on a later page, which matches.
Remember
- repeated root
Examples
- (x+3)2=0
- x=-3, -3
- x2-6x+9=(x-3)2
Remember
- x^2 - k^2
- (x+k)^2
- x = -3 repeated
- middle term
- +/- roots
- repeated root
- x^2 - k^2
- (x+k)^2
How the ideas connect
Worked: solve x^2 + 6x + 9 = 0. x^2 + 6x + 9 = 0 (x + 3)(x + 3) = 0 (x + 3)^2 = 0 x + 3 = 0 x = -3 (repeated)
Deeper understanding
Go beyond memorising Special Factors. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: solve x^2 + 6x + 9 = 0. x^2 + 6x + 9 = 0 (x + 3)(x + 3) = 0 (x + 3)^2 = 0 x + 3 = 0 x = -3 (repeated)
Common mistakes
Exam focus
Core Idea
DIFFERENCE OF SQUARES
PERFECT SQUARE
Worked Method
Exam focus
Quick revision
- x^2 - k^2
- (x+k)^2
- x = -3 repeated
- middle term
- SPECIAL FACTORS: Difference of squares: x^2 - 9 = (x-3)(x+3), so x = +/- 3. Perfect square: x^2 + 6x + 9 = (x+3)^2, a
- DIFFERENCE OF SQUARES: x^2 - 9 = 0 -> (x-3)(x+3)=0 -> x=3 or x=-3. Same idea for 4x^2 - 25 after you see (2x-5)(2x+5).
x^2 - 9 is not (x-3)^2. (x-3)^2 = x^2 - 6x + 9. The middle term tells you.
For long answers on Special Factors: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- x^2 - k^2
- (x+k)^2
- x = -3 repeated
- middle term
Important formulas
- x^2 - k^2
- (x+k)^2
- x = -3 repeated
Common exam traps
- x^2 - 9 is not (x-3)^2. (x-3)^2 = x^2 - 6x + 9. The middle term tells you.
- Not (x-3)^2
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
x^2 - 9 is not (x-3)^2. (x-3)^2 = x^2 - 6x + 9. The middle term tells you.
- Mistake
Not (x-3)^2
- Mistake
Do not treat DIFFERENCE OF SQUARES and PERFECT SQUARE as the same step; each has a different purpose.
- Mistake
x^2 - 9 is not (x-3)^2. (x-3)^2 = x^2 - 6x + 9. The middle term tells you.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
x^2 - k^2
(x+k)^2
x = -3 repeated
middle term
Difference of squares: x^2 - 9 = (x-3)(x+3), so x = +/- 3. Perfect square: x^2 + 6x + 9 = (x+3)^2, a repeated root x=-3. These are still quadratics; they just f
x = -3 repeated
- Definition
- Difference of squares: x^2 - 9 = (x-3)(x+3), so x = +/- 3. Perfect square: x^2 + 6x + 9 = (x+3)^2, a repeated root x=-3. These are still quadratics; they just f
- Formula
- x = -3 repeated
- Example
- x2-9=0 -> +/-3
- Critical warning
- x^2 - 9 is not (x-3)^2. (x-3)^2 = x^2 - 6x + 9. The middle term tells you.
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Exam Practice
Define
Q1. Define Special Factors.
Explain
Q2. Explain: DIFFERENCE OF SQUARES.
Example
Q3. Give one clear example of Special Factors and say why it fits.
Mistake
Q4. State one common mistake students make about Special Factors.
Exam note
Q5. Write a short exam note on Special Factors: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.