Algebraic expressions and formulae give you the algebraic toolkit you will lean on for the rest of secondary school. Quadratic equations, graphs and even Additional Math all start with confidently expanding, factorising and simplifying expressions.
At Sec 2 this covers expanding two linear expressions, the three special identities, factorising ax² + bx + c by splitting the middle term, and multiplying and dividing algebraic fractions. G3 students also meet grouping factorisation, addition and subtraction of algebraic fractions, general expansion and changing the subject of a formula at this level.
What these notes cover
- •Expanding two linear expressions
- •Special algebraic identities
- •Using identities for mental arithmetic
- •Factorising ax² + bx + c by splitting the middle term
- •Choosing a factorising method
- •Multiplying and dividing algebraic fractions
- •Common mistakes
- •Note for G3 stream students
- •Summary
Key definitions
- Perfect square
- An expansion of the form (a + b)² = a² + 2ab + b² or (a − b)² = a² − 2ab + b², which should be recognised on sight rather than re-expanded from scratch.
- Difference of two squares
- An expression of the form a² − b², which factorises as (a + b)(a − b). The sum a² + b² does not factorise over real numbers using this identity.
- Splitting the middle term
- Rewriting bx in ax² + bx + c as mx + nx, where m × n = ac and m + n = b, then factorising the resulting four-term expression by grouping.
- Cancelling factors
- In an algebraic fraction, you can only cancel factors that are common to the whole numerator and the whole denominator, never individual terms.
- Reciprocal
- The second fraction flipped over: dividing by (C ÷ D) is the same as multiplying by (D ÷ C).
Key formulas
Expansion of two linear expressions
(ax + b)(cx + d) = acx² + (ad + bc)x + bd
Every term in the first bracket must multiply every term in the second bracket.
Perfect square (sum)
(a + b)² = a² + 2ab + b²
Also shortcuts squaring numbers close to a round number, e.g. 105² = (100 + 5)².
Perfect square (difference)
(a − b)² = a² − 2ab + b²
The middle term is −2ab. Recognise the pattern on sight instead of re-expanding.
Difference of two squares
a² − b² = (a + b)(a − b)
Can be used in reverse for mental arithmetic, e.g. 98 × 102 = (100 − 2)(100 + 2). Only a² − b² factorises this way, not a² + b².
Multiplying algebraic fractions
Factorise every numerator and denominator completely, then cancel common factors before multiplying across.
Dividing algebraic fractions
Flip the second fraction (multiply by its reciprocal), then proceed exactly as for multiplication.
Worked example
Factorise completely 6x² − 5x − 6.
- 1Here a = 6, b = −5 and c = −6, so ac = 6 × (−6) = −36.
- 2Find two numbers that multiply to −36 and add to −5: these are −9 and 4.
- 3Split the middle term: 6x² − 5x − 6 = 6x² − 9x + 4x − 6.
- 4Factorise by grouping: 6x² − 9x + 4x − 6 = 3x(2x − 3) + 2(2x − 3).
- 5Take out the common bracket: 3x(2x − 3) + 2(2x − 3) = (2x − 3)(3x + 2).
- 6Check by expanding back: (2x − 3)(3x + 2) = 6x² + 4x − 9x − 6 = 6x² − 5x − 6. ✓
Watch out: Sign slips are easy when the constant term is negative. Always double-check that the two numbers chosen multiply to give ac and add to give b.
Common mistakes
Not distributing a negative sign across the whole bracket
Wrong: −(x + 3) = −x + 3
Right: −(x + 3) = −x − 3. The negative sign multiplies every term inside the bracket.
Cancelling terms instead of factors
Wrong: Cancelling the x in (x + 3) ÷ x.
Right: Factorise fully first. Only cancel factors that are common to the whole numerator and the whole denominator.
Confusing a² − b² with a² + b²
Wrong: Treating a² + b² as if it factorises like a² − b².
Right: Only a² − b² = (a + b)(a − b). a² + b² does not factorise over real numbers using this identity.
Questions students ask
How do you factorise ax² + bx + c in Sec 2 Math?
Always try a common factor first, then look for a difference-of-two-squares or perfect-square pattern. If neither applies, find m and n with m × n = ac and m + n = b, rewrite bx as mx + nx, and factorise by grouping. Check the answer by expanding it back out.
What are the three algebraic identities in Sec 2 Math?
They are (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and a² − b² = (a + b)(a − b). Recognising them on sight saves re-expanding and shortcuts mental arithmetic, e.g. 98 × 102 = 100² − 2² = 9996.
How do you divide algebraic fractions?
Flip the second fraction and multiply by its reciprocal, then proceed exactly as for multiplication. Factorise everything completely first, and cancel common factors before multiplying across.
Do G2 and G3 students learn the same algebra in Sec 2?
Both streams cover expansion of two linear expressions, the identities, factorising ax² + bx + c, and multiplying and dividing algebraic fractions. G3 students also meet grouping factorisation, addition and subtraction of algebraic fractions, general expansion of three or more brackets, and changing the subject of a formula at Sec 2, while G2 students meet these in Sec 3/4.
In the full PDF
- •Worked example: expanding and simplifying (2x − 3)(x + 5) − x(2x − 7)
- •Worked examples using identities for mental arithmetic, including 98 × 102 and 105² without a calculator
- •Worked example on simplifying a division of algebraic fractions by factorising and cancelling
- •The remaining common mistakes, including skipping the common-factor check
- •End-of-topic summary checklist