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Gaokao Math Formulas

102 cards · 🇺🇸 English

Essential mathematical formulas for the Chinese National College Entrance Examination (Gaokao)

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Quadratic formula

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  1. 1Quadratic formula
    For ax² + bx + c = 0: x = (-b ± √(b² - 4ac)) / 2a
  2. 2Discriminant of quadratic
    Δ = b² - 4ac; If Δ > 0: two real roots; Δ = 0: one root; Δ < 0: no real roots
  3. 3Sum and product of roots
    For ax² + bx + c = 0: sum = -b/a, product = c/a
  4. 4Binomial expansion
    (a + b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏ × bᵏ where C(n,k) = n!/(k!(n-k)!)
  5. 5Difference of squares
    a² - b² = (a + b)(a - b)
  6. 6Perfect square formulas
    (a + b)² = a² + 2ab + b²; (a - b)² = a² - 2ab + b²
  7. 7Sum of cubes
    a³ + b³ = (a + b)(a² - ab + b²)
  8. 8Difference of cubes
    a³ - b³ = (a - b)(a² + ab + b²)
  9. 9Arithmetic sequence general term
    aₙ = a₁ + (n - 1)d where d is common difference
  10. 10Arithmetic sequence sum
    Sₙ = n(a₁ + aₙ)/2 = n(2a₁ + (n-1)d)/2
  11. 11Geometric sequence general term
    aₙ = a₁ × rⁿ⁻¹ where r is common ratio
  12. 12Geometric sequence sum (finite)
    Sₙ = a₁(1 - rⁿ)/(1 - r) for r ≠ 1; Sₙ = na₁ for r = 1
  13. 13Geometric sequence sum (infinite)
    S = a₁/(1 - r) for |r| < 1
  14. 14Sine rule
    a/sin A = b/sin B = c/sin C = 2R (R is circumradius)
  15. 15Cosine rule
    a² = b² + c² - 2bc cos A; or cos A = (b² + c² - a²)/2bc
  16. 16Triangle area (sine)
    Area = (1/2)ab sin C
  17. 17Triangle area (Heron's formula)
    Area = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2
  18. 18Pythagorean identity
    sin²θ + cos²θ = 1
  19. 19Tangent identity
    tan θ = sin θ / cos θ
  20. 20Reciprocal trigonometric identities
    csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ
  21. 21Double angle: sine
    sin 2θ = 2 sin θ cos θ
  22. 22Double angle: cosine
    cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
  23. 23Double angle: tangent
    tan 2θ = 2tan θ / (1 - tan²θ)
  24. 24Half angle: sine
    sin(θ/2) = ±√((1 - cos θ)/2)
  25. 25Half angle: cosine
    cos(θ/2) = ±√((1 + cos θ)/2)
  26. 26Sum of angles: sine
    sin(α + β) = sin α cos β + cos α sin β
  27. 27Sum of angles: cosine
    cos(α + β) = cos α cos β - sin α sin β
  28. 28Difference of angles: sine
    sin(α - β) = sin α cos β - cos α sin β
  29. 29Difference of angles: cosine
    cos(α - β) = cos α cos β + sin α sin β
  30. 30Product-to-sum: sin α sin β
    sin α sin β = (1/2)[cos(α - β) - cos(α + β)]

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