Quadratic Formula: x = (-b ± √(b²-4ac)) / (2a)
Discriminant Δ = b²-4ac
Δ > 0 → 2 real roots | Δ = 0 → 1 equal root | Δ < 0 → no real roots
Forms: y = ax²+bx+c (standard)
y = a(x-h)²+k (vertex)
y = a(x-p)(x-q) (factored)
Example: 2x²-5x-3=0 → Δ=25+24=49 → x=(5±7)/4 → x=3 or x=-1/2
Linear: y = mx + c gradient m = (y₂-y₁)/(x₂-x₁)
Parallel: m₁=m₂ | Perpendicular: m₁×m₂=-1
Vertex: x = -b/(2a) (axis of symmetry)
Inverse: swap x and y, solve for y
Example: Vertex of y=2x²-8x+5: x=-(-8)/(4)=2, y=8-16+5=-3 → vertex (2,-3)
SOH-CAH-TOA (right triangles):
sinθ=opp/hyp cosθ=adj/hyp tanθ=opp/adj
Sine Rule: a/sinA = b/sinB = c/sinC
Cosine Rule: c² = a²+b²-2ab·cosC
cosC = (a²+b²-c²)/(2ab)
Area: A = ½ab·sinC
Exact values: sin30°=½, cos60°=½, tan45°=1, sin45°=√2/2, sin60°=√3/2
Mean: x̄ = Σx/n (or Σfx/Σf for grouped)
Median: middle value (ordered)
IQR: Q3 - Q1
Population SD: σ = √(Σ(x-x̄)²/n)
Sample SD: s = √(Σ(x-x̄)²/(n-1))
Example: {3,5,7,7,9}: mean=31/5=6.2, median=7, mode=7, range=6
Circle: A=πr², C=2πr
Arc length: L = (θ/360)×2πr
Sector area: A = (θ/360)×πr²
Cone: V=⅓πr²h, SA=πrl+πr²
Sphere: V=⅔πr³, SA=4πr²
Space diagonal: d=√(l²+w²+h²)
Example: Cone r=6, l=10: SA=π(6)(10)+π(36)=60π+36π=96π cm²
Arithmetic (AP): uₙ=a+(n-1)d
Sₙ = n/2×(2a+(n-1)d) = n/2×(a+l)
Geometric (GP): uₙ=ar^(n-1)
Sₙ = a(rⁿ-1)/(r-1) [r≠1]
S∞ = a/(1-r) [|r|<1]
Example: AP a=4, d=3, S₁₀=10/2×(8+27)=5×35=175
P(A) = favourable/total
P(A∪B) = P(A)+P(B)-P(A∩B)
P(A|B) = P(A∩B)/P(B)
Independent: P(A∩B) = P(A)×P(B)
Complement: P(A') = 1-P(A)
Example: P(A)=0.5,P(B)=0.4,P(A∩B)=0.2: P(A)×P(B)=0.2 → independent; P(A|B)=0.5
aᵐ×aᵑ=a^(m+n) | aᵐ/aᵑ=a^(m-n)
(aᵐ)ᵑ=a^(mn) | a⁰=1 | a^(-n)=1/aᵑ
a^(1/n)=ⁿ√a
Surds: √a×√b=√(ab)
Rationalise: ×(√a-√b)/(√a-√b)
Example: 6/(√5-√2)×(√5+√2)/(√5+√2) = 6(√5+√2)/3 = 2(√5+√2)