MathematicsChapter 2: Quadratic Equations
Special Factors
Difference of squares: x^2 - 9 = (x-3)(x+3), so x = +/- 3.
Learning objectives
- 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.
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 factor quickly.
Mathematical form
x = -3 (repeated)
x = +/- 3. Perfect square
x=-3. These are still quadratics
Where / symbols
- 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.
Overview
Two patterns to spot:
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).
Examples x^2-9=0 -> +/-3 · x^2-16=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.
Examples (x+3)^2=0 · x=-3, -3 · x^2-6x+9=(x-3)^2
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.
Worked example
- solve x2 + 6x + 9 = 0.
- x2 + 6x + 9 = 0
- (x + 3)(x + 3) = 0
- (x + 3)2 = 0
- x + 3 = 0
- x = -3 (repeated)
In simple words
x2 - 9 is not (x-3)2. (x-3)2 = x2 - 6x + 9. The middle term tells you.
Key points
- x^2 - k^2
- (x+k)^2
- +/- roots
- repeated root
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Special Factors
Sticky Notes · Quick Revision
Core Idea
- 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.
Worked Method
- Worked: solve x2 + 6x + 9 = 0.
- x2 + 6x + 9 = 0
- (x + 3)(x + 3) = 0
- (x + 3)2 = 0
- x + 3 = 0
- x = -3 (repeated)
Common Misconception
- x^2 - 9 is not (x-3)^2. (x-3)^2 = x^2 - 6x + 9. The middle term tells you.
FBISE Exam Check
- x2 - 9 is not (x-3)2. (x-3)2 = x2 - 6x + 9. The middle term tells you.
60-second revision
- 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)
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