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ALGEBRA

Exam Cog
①NUMBER SETS

N — Naturals (> 0)

Z — Integers (+/− whole numbers)

Q — Rationals (fractions)

R — Reals (any number)

R−Q — Irrationals

②NOTATION & PERFECT SQUARES

In 3x2 + 7x + 9:

• 3 = coefficient   • x = variable

• 2 = power / degree   • 9 = constant

(a + b)2 = a2 + 2ab + b2

(a − b)2 = a2 − 2ab + b2

★ Square the 1st, square the 2nd, then 1st × 2nd doubled
③FACTORISING — 5 TYPES
  1. Grouping (2 or 4 terms): take out the common factor
  2. Quadratic trinomial: guide-number / double brackets / big X
  3. Difference of two squares:
    a2 − b2 = (a − b)(a + b)
  4. Sum / diff of cubes:
    a3 + b3 = (a + b)(a2 − ab + b2)
    a3 − b3 = (a − b)(a2 + ab + b2)
  5. Combinations: take out the common factor first
④QUADRATIC EQN   ax2+bx+c=0

• Factorise & let each factor = 0,  OR

• x = [ −b ± √(b2 − 4ac) ] / 2a   [tables]

Discriminant   b2 − 4ac:

> 0 → 2 real roots

< 0 → complex (unreal) roots

= 0 → equal roots (perfect square)

≥ 0 → real roots

★ Complete the square: halve the coeff of x, then square it
⑤FRACTIONS & SIMULTANEOUS EQNS

Algebraic fractions:

Add / Sub / Equality → common denominator. Complex: simplify top & bottom to single fractions, then flip & multiply.

Simultaneous equations:

  1. Two lines (basic)
  2. Line + curve: from the linear get x= or y=, sub into the curve
  3. Substitution — e.g. let a = 1/x, b = 1/y
  4. 3×3 system — multiply the WHOLE line, watch signs
  5. Identity: equate coeffs of x & the constants on both sides
⑥ROOTS & FACTOR THEOREM

Roots of a quadratic (coeff of x2 = 1):

x2 − (sum of roots)x + (product) = 0

Factor Theorem: if f(k) = 0 then (x − k) is a factor

Methods: sub in  OR  divide in

Types: find factors (try ±1, ±2 …); 1 unknown (sub root, = 0); 2 unknowns; variable factor (quadratic × linear (x + k))

⑦INEQUALITIES & |x|

|x|: always ≥ 0. Methods: (1) split ± cases (2) square both sides (3) graph it

Inequalities: treat like equations BUT flip the sign when you × or ÷ by a negative

Quadratic: solve = 0; smaller root, x, larger root — sketch the graph for the sign

With fractions: × by the denominator SQUARED (always +, so the sign is safe)

With |·|: square both sides

★ Proofs: perfect squares backwards, or use the given info (a > b ⇒ a − b > 0)
⑧SURDS

√(ab) = √a · √b    √(a/b) = √a / √b

(√a)2 = a    √a · √a = a

Simplify: factor out the perfect square

Add / sub: only the same irrational part

Check answers: a positive base ⇏ a negative

⚠ WATCH: √(a + b) ≠ √a + √b
⑨INDICES

Rules are in the tables. To solve: get the same base, then equate the powers.

Quadratic in indices: e.g. 32x+1 = 3·(3x)2 → let t = 3x

⚠ WATCH: 2(2x) ≠ 4x — it's 2x+1
⚠ WATCH: 5·3n ≠ 15n (very common slip!)
⑩LOGS

Definition:

logb(n) = p  ⇔  bp = n

★ BNP — Base, to the Power p, equals the Number

Rules:

log(ab) = log a + log b

log(a/b) = log a − log b

log an = n · log a

⚠ WATCH: (log a)n ≠ n·log a  ;  log a / log b ≠ log(a − b)