Quadratic Expressions
Learn to expand brackets, factorise quadratics, and apply the difference of two squares identity to simplify algebraic expressions.
Expanding Quadratic Expressions
A quadratic expression has the form ax2 + bx + c, where a ≠ 0. We expand quadratics by multiplying every term in one bracket by every term in the other using the FOIL method: First, Outer, Inner, Last.
FOIL Method
(x + a)(x + b) = x2 + bx + ax + ab = x2 + (a + b)x + ab
Example: (x + 3)(x + 5)
First: x × x = x2
Outer: x × 5 = 5x
Inner: 3 × x = 3x
Last: 3 × 5 = 15
= x2 + 8x + 15
Example: (2x − 1)(x + 4)
First: 2x × x = 2x2
Outer: 2x × 4 = 8x
Inner: −1 × x = −x
Last: −1 × 4 = −4
= 2x2 + 7x − 4
Factorising Monic Quadratics
To factorise x2 + bx + c, find two numbers that multiply to c and add to b. These numbers become the constants in two brackets. This is the reverse of expanding.
The Rule
x2 + bx + c = (x + p)(x + q) where p × q = c and p + q = b
Factorise x2 + 7x + 12
Step 1: Find two numbers that multiply to 12 and add to 7.
Step 2: Consider pairs: 1 × 12 = 12 (sum 13), 2 × 6 = 12 (sum 8), 3 × 4 = 12 (sum 7). ✓
Answer: x2 + 7x + 12 = (x + 3)(x + 4)
Factorise x2 − 2x − 15
Step 1: Need two numbers that multiply to −15 and add to −2.
Step 2: 3 × (−5) = −15 and 3 + (−5) = −2. ✓
Answer: x2 − 2x − 15 = (x + 3)(x − 5)
Difference of Two Squares
The difference of two squares is a special identity for expressions of the form a2 − b2. It arises from expanding the conjugate pair (a + b)(a − b) where the middle terms cancel.
Identity
a2 − b2 = (a + b)(a − b)
Factorise x2 − 25
= x2 − 52
= (x + 5)(x − 5)
Factorise 4x2 − 49
= (2x)2 − 72
= (2x + 7)(2x − 7)
Remember: Always look for a common factor first. The sum a2 + b2 does not factorise over the real numbers.
Key Vocabulary
Quadratic expression
A polynomial of degree 2, written as ax2 + bx + c where a ≠ 0.
Expanding
Removing brackets by multiplying each term, using FOIL for two binomials.
Factorising
Writing an expression as a product of simpler factors; the reverse of expanding.
Difference of two squares
The identity a2 − b2 = (a + b)(a − b).
Worked Examples
Expand and simplify (x + 6)2
Step 1: Write as (x + 6)(x + 6).
Step 2: Apply FOIL: x2 + 6x + 6x + 36
Step 3: Collect like terms: x2 + 12x + 36
This illustrates (a + b)2 = a2 + 2ab + b2.
Factorise x2 − 5x + 6
Step 1: Need two numbers that multiply to +6 and add to −5.
Step 2: (−2) × (−3) = 6 and (−2) + (−3) = −5. ✓
Answer: (x − 2)(x − 3)
Check: (x − 2)(x − 3) = x2 − 5x + 6 ✓
Factorise fully: 3x2 − 27
Step 1: Take out the common factor of 3: 3(x2 − 9)
Step 2: Recognise x2 − 9 = x2 − 32 (difference of two squares).
Answer: 3(x + 3)(x − 3)
Knowledge Check
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Key Concepts Summary
- ●A quadratic expression has the form ax2 + bx + c where a ≠ 0.
- ●Use FOIL to expand two brackets: First, Outer, Inner, Last — then collect like terms.
- ●To factorise x2 + bx + c, find two numbers that multiply to c and add to b.
- ●Difference of two squares: a2 − b2 = (a + b)(a − b).
- ●Always look for a common factor first before applying other factorising techniques.