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Grade 11 · Equations
The Discriminant & Nature of Roots: Full Guide
MEMORANDUM
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Date: 
Mark
  / 20

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

  1. 1
    Δ for x2 − 6x + 9 = 0, and the nature of roots. (classify)
    (2)
    36 − 36 = 0 → one real (equal) root✓✓ (2)
  2. 2
    Δ for x2 + 2x + 5 = 0, and the nature of roots. (classify)
    (2)
    4 − 20 = −16 < 0 → no real roots✓✓ (2)
  3. 3
    Δ for 2x2 − 7x + 3 = 0, and the nature. (classify)
    (2)
    49 − 24 = 25 > 0 → two real roots✓✓ (2)
  4. 4
    Are the roots of x2 − 4x + 4 = 0 rational? (rational?)
    (2)
    Δ = 0 (perfect square) → rational (equal)✓✓ (2)
  5. 5
    Rational or irrational: x2 − 2x − 2 = 0? (rational?)
    (2)
    Δ = 4 + 8 = 12, not a perfect square → irrational✓✓ (2)
  6. 6
    For which k does x2 + kx + 4 = 0 have equal roots? (equal, find k)
    (2)
    k2 − 16 = 0 → k = ±4✓✓ (2)
  7. 7
    For which k does x2 − 6x + k = 0 have equal roots? (equal, find k)
    (2)
    36 − 4k = 0 → k = 9✓✓ (2)
  8. 8
    For which k does x2 + 2x + k = 0 have two real roots? (real, find k)
    (2)
    4 − 4k > 0 → k < 1✓✓ (2)
  9. 9
    For which k does x2 − 2x + k = 0 have no real roots? (non-real, find k)
    (2)
    4 − 4k < 0 → k > 1✓✓ (2)
  10. 10
    Δ for 3x2 − 5x + 1 = 0. (compute)
    (2)
    25 − 12 = 13 (two real, irrational)✓✓ (2)
TOTAL: 20 marks
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