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TMUA 2023 · Paper 2 · Question 18 of 20

TMUA 2023 Paper 2 Question 18

Algebra and functions — Quartics · four distinct real roots. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Algebra and functionsQuartics · four distinct real roots6 options
The equation x4 + bx2 + c = 0 has four distinct real roots if and only if which of the following conditions is satisfied?
  1. Ab2 > 4c
  2. Bb2 < 4c
  3. Cc > 0 and b > 2√c
  4. Dc > 0 and b < −2√c
  5. Ec < 0 and b < 0
  6. Fc < 0 and b > 0
Show the answer and worked solution
answer · D
  1. Ab2 > 4c
  2. Bb2 < 4c
  3. Cc > 0 and b > 2√c
  4. Dc > 0 and b < −2√c
  5. Ec < 0 and b < 0
  6. Fc < 0 and b > 0
Put u = x2, so u2 + bu + c = 0. Each strictly positive root u gives two real values x = ±√ u, so four distinct real roots need two distinct positive roots in u. That needs a positive discriminant, b2 > 4c; a positive sum of roots, −b > 0, so b < 0; and a positive product, c > 0. With c > 0 and b < 0, the discriminant condition reads b < −2√c.