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TMUA 2020 · Paper 1 · Question 20 of 20

TMUA 2020 Paper 1 Question 20

Algebra and functions — Repeated roots · counting parameter values. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 1Algebra and functionsRepeated roots · counting parameter values6 optionshard
For how many values of a is the equation (x − a)(x2 − x + a) = 0 satisfied by exactly two distinct values of x?
  1. A0
  2. B1
  3. C2
  4. D3
  5. E4
  6. Fmore than 4
Show the answer and worked solution
answer · C
  1. A0
  2. B1
  3. C2
  4. D3
  5. E4
  6. Fmore than 4
One root is always x = a; the rest come from x2 − x + a = 0, whose discriminant is 1 − 4a. If 1 − 4a < 0 there is only the single root x = a, so exactly two distinct roots can happen in only two ways. Either the quadratic has a repeated root distinct from a: 1 − 4a = 0 gives a = 14, with repeated root x = 12 ≠ 14, so the roots are 14 and 12 — that works. Or the quadratic has two distinct roots, one of which is a: substituting x = a gives a2 − a + a = a2 = 0, so a = 0, and then the roots are 0 and 1 — that works too. That is 2 values of a.