Algebra: Expressions and Equations

Algebra is the language of mathematics. It allows us to generalise, describe patterns, and solve problems using letters to represent unknown or variable quantities.

---

Algebraic Notation

Notation	Meaning
ab	a × b (no multiplication sign)
3y	3 × y
a²	a × a
a/b	a ÷ b
(a + b)	a and b treated as a single group

Terms: parts of an expression separated by + or −. In 3x² − 2x + 5, the terms are 3x², −2x, and 5. Coefficient: the number in front of a variable. In 3x², the coefficient is 3. Constant: a number with no variable. In 3x² − 2x + 5, the constant is 5.

---

Substitution

Replace each letter with its given value and calculate using BIDMAS.

Example: If a = 3 and b = −2, find 4a² − 3b. = 4(3²) − 3(−2) = 4(9) + 6 = 36 + 6 = 42

---

Simplifying Expressions — Collecting Like Terms

Like terms have identical letter parts (including powers).

Like terms	Unlike terms
5x and 3x → 8x	5x and 3x² (different powers)
2ab and −7ab → −5ab	2ab and 2a (different variables)

Example: Simplify 3x² + 5x − x² + 2 − 4x = (3x² − x²) + (5x − 4x) + 2 = 2x² + x + 2

---

Expanding Brackets

Single bracket: multiply each term inside by the term outside. 5(3x − 2) = 15x − 10

Double brackets (FOIL): multiply each term in the first bracket by each term in the second. (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15

Special cases:

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• (a + b)(a − b) = a² − b² (difference of two squares)

---

Factorising

Common factor: the reverse of expanding. Find the HCF and take it outside. 12x² − 8x = 4x(3x − 2)

Factorising quadratics (x² + bx + c): find two numbers that multiply to give c and add to give b. x² + 7x + 12: need numbers → ×12 and +7 → 3 and 4 → (x + 3)(x + 4)