| formula 📄 GIVEN | x = [−b ± √(b² − 4ac)]/2a |
| complete sq. | a(x − h)² + k → TP (h, k) |
| always +ve | a > 0, D < 0 |
| always −ve | a < 0, D < 0 |
| D > 0 | 2 real roots / line cuts at 2 pts |
| D = 0 | equal roots / tangent |
| D < 0 | no real roots / no intersection |
| D ≥ 0 | real roots / meets |
| (x − a)(x − b) < 0 | a < x < b (between) |
| (x − a)(x − b) > 0 | x < a or x > b (outside) |
| √a√b | √(ab) |
| rationalise | × conjugate a ∓ √b |
| (√a + √b)(√a − √b) | a − b |
| surd equation | square, then check |
| remainder | f(x) ÷ (x − a) → f(a) |
| (ax − b) | remainder f(b/a) |
| factor | f(a) = 0 ⇔ (x − a) factor |
| a³ + b³ | (a + b)(a² − ab + b²) |
| a³ − b³ | (a − b)(a² + ab + b²) |
| (ax+b)(cx+d) | A/( ) + B/( ) |
| (ax+b)(cx+d)² | A/( ) + B/(cx+d) + C/(cx+d)² |
| (ax+b)(x²+c²) | A/(ax+b) + (Bx+C)/(x²+c²) |
| improper | divide first |
| (a + b)ⁿ | Σ ⁿCᵣ aⁿ⁻ʳ bʳ |
| general term | Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ |
| ⁿCᵣ | n!/(r!(n − r)!) |
| find term | set power of x, solve r |
| y = aˣ | x = logₐ y |
| log xy | log x + log y |
| log x/y | log x − log y |
| log xⁿ | n log x |
| change base | logₐ b = lg b / lg a |
| logₐ a · logₐ 1 | 1 · 0 |
| sin 30, 45, 60 | ½, √2/2, √3/2 |
| cos 30, 45, 60 | √3/2, √2/2, ½ |
| tan 30, 45, 60 | √3/3, 1, √3 |
| π/6, π/4, π/3 | 30°, 45°, 60° |
| Q1 A | all + |
| Q2 S | sin + (180 − α) |
| Q3 T | tan + (180 + α) |
| Q4 C | cos + (360 − α) |
| sin⁻¹, tan⁻¹ | −90° to 90° |
| cos⁻¹ | 0° to 180° |
| sin² + cos² | 1 |
| sec²A | 1 + tan²A |
| cosec²A | 1 + cot²A |
| sin(A ± B) | sAcB ± cAsB |
| cos(A ± B) | cAcB ∓ sAsB |
| sin 2A | 2 sA cA |
| cos 2A | c² − s² = 2c² − 1 = 1 − 2s² |
| tan 2A | 2t/(1 − t²) |
| tan | sin/cos |
| sec, cosec, cot | 1/cos, 1/sin, 1/tan |
| R-formula | a sin θ + b cos θ = R sin(θ + α) |
| R, α | √(a² + b²), tan α = b/a |
| y = a sin bx + c | amp |a|, period 360°/b |
| y = a sin(x/b) + c | period 360° × b |
| y = a tan bx | period 180°/b |
| parallel | m₁ = m₂ |
| perpendicular | m₁m₂ = −1 |
| midpoint | ((x₁+x₂)/2, (y₁+y₂)/2) |
| area | shoelace ½|Σx₁y₂ − Σy₁x₂| |
| standard | (x − a)² + (y − b)² = r² |
| general | x² + y² + 2gx + 2fy + c = 0 |
| centre | (−g, −f) |
| radius | √(g² + f² − c) |
| tangent | ⊥ radius |
| y = axⁿ | lg y = n lg x + lg a |
| y = kbˣ | lg y = x lg b + lg k |
| Y, X | no unknown constants inside |
| c | Y-intercept only if X starts at 0 |
| alternate segment | ∠ between tangent & chord = ∠ in alt. segment |
| midpoint theorem | MN ∥ BC, MN = ½BC |
| parallel lines | alt. / corr. / int. ∠s |
| congruent | SSS SAS ASA RHS |
| similar | AA SSS SAS |
| semicircle | 90° |
| centre | 2 × circumference |
| same segment | equal |
| cyclic quad | opp. ∠s 180° |
| tangents from pt | equal |
| tangent | ⊥ radius |
| y | dy/dx | y | dy/dx |
|---|---|---|---|
| xⁿ | nxⁿ⁻¹ | sin x | cos x |
| (ax + b)ⁿ | an(ax + b)ⁿ⁻¹ | cos x | −sin x |
| eᵃˣ⁺ᵇ | aeᵃˣ⁺ᵇ | tan x | sec²x |
| ln(ax + b) | a/(ax + b) | sin(ax + b) | a cos(ax + b) |
| chain | dy/du × du/dx |
| product | uv′ + vu′ |
| quotient | (vu′ − uv′)/v² |
| angles | radians |
| stationary | dy/dx = 0 |
| d²y/dx² > 0 | minimum |
| d²y/dx² < 0 | maximum |
| increasing | dy/dx > 0 |
| normal gradient | −1/m |
| connected rates | dA/dt = dA/dx × dx/dt |
| ∫ | Result | ∫ | Result |
|---|---|---|---|
| xⁿ | xⁿ⁺¹/(n + 1) + c | sin(ax + b) | −(1/a)cos(ax + b) + c |
| (ax + b)ⁿ | (ax + b)ⁿ⁺¹/[a(n + 1)] + c | cos(ax + b) | (1/a)sin(ax + b) + c |
| eᵃˣ⁺ᵇ | (1/a)eᵃˣ⁺ᵇ + c | sec²x | tan x + c |
| ∫ₐᵇ f(x) dx | area above x-axis |
| below axis | negative → take | | |
| crosses axis | split at roots |
| with y-axis | ∫ x dy |
| not tested | area between 2 curves |
| v | ds/dt |
| a | dv/dt |
| s | ∫ v dt |
| v | ∫ a dt |
| at rest | v = 0 |
| max v | a = 0 |
| distance | split where v = 0 |
| exact answers | leave surds, π, ln |
| trig equations | list all in interval |
| logs / surds | reject invalid roots |
| "show that" | end with given result |
| "hence" | use the previous part |