Factorise quadratics: special cases
Key Notes:
Understanding Quadratic Expressions
- A quadratic expression is in the form ax² + bx + c.
- Special cases involve patterns that make factorisation easier.
Perfect Square Trinomials
- These take the form a² + 2ab + b² = (a + b)² or a² – 2ab + b² = (a – b)².
- Example: x² + 6x + 9 = (x + 3)(x + 3) = (x + 3)².
Difference of Squares
- This follows the identity a² – b² = (a – b)(a + b).
- Example: x² – 9 = (x – 3)(x + 3).
Common Factor Method
- Always check for common factors before applying special cases.
- Example: 4x² – 36 = 4(x² – 9) = 4(x – 3)(x + 3).
Zero Product Property
- If a quadratic expression is factorised and set to zero, solve by setting each factor to zero.
- Example: (x – 5)(x + 2) = 0 gives x = 5 or x = -2.
Recognizing Special Cases
- Identify perfect squares and differences of squares quickly.
- Look for missing middle terms (b = 0) in the difference of squares.
Practice with Different Forms
- Example 1: x² + 16x + 64 = (x + 8)².
- Example 2: 49y² – 25 = (7y – 5)(7y + 5).
Application in Problem Solving
- Use factorisation to simplify algebraic expressions.
- Apply in solving quadratic equations efficiently.
Factorising perfect square trinomials:
a2+2ab+b2=(a+b)2
a2–2ab+b2=(a–b)2
Learn with an example
🥎 Factorise.
c2–8c+16=——
Notice that c2–8c+16 is a perfect square trinomial because it can be written in the form a2–2ab+b2, where a is c and b is 4.
a2–2ab+b2
c2–2c4+42
c2–8c+16
Now use the formula for factorising perfect square trinomials.
a2–2ab+b2=(a–b)2
c2–2c4+42=(c–4)2
c2–8c+16=(c–4)2
The factorised form of c2–8c+16 is (c–4)2.
Finally, check your work.
(c–4)2
(c–4)(c–4)Expand
c2–4c–4c+16Apply the distributive property (FOIL)
c2–8c+16
Yes, c2–8c+16=(c–4)2.
🥎 Factorise.
9x2–6x+1=——
Notice that 9x2–6x+1 is a perfect square trinomial because it can be written in the form a2–2ab+b2, where a is 3x and b is 1.
a2–2ab+b2
(3x)2–2 . 3x . 1+12
9x2–6x+1
Now use the formula for factorising perfect square trinomials.
a2–2ab+b2=(a–b)2
(3x)2–2 . 3x . 1+12=(3x–1)2
9x2–6x+1=(3x–1)2
The factorised form of 9x2–6x+1 is (3x–1)2.
Finally, check your work.
(3x–1)2
(3x–1)(3x–1)Expand
9x2–3x–3x+1Apply the distributive property (FOIL)
9x2–6x+1
Yes, 9x2–6x+1=(3x–1)2.
🥎 Factorise.
z2–1=—–
Notice that z2–1 is a difference of squares, because it can be written in the form a2–b2, where a is z and b is 1.
a2–b2
z2–12
z2–1
Now use the formula for factorising a difference of squares.
a2–b2=(a+b)(a–b)
z2–12=(z+1)(z–1)
z2–1=(z+1)(z–1)
The factorised form of z2–1 is (z+1)(z–1).
Finally, check your work.
(z+1)(z–1)
z2+z–z–1Apply the distributive property (FOIL)
z2–1
Yes, z2–1=(z+1)(z–1).
let’s practice!