Multiple choice

Suppose that $m$ and $n$ are integers such that both the quadratic equations $x^2 + mx - n = 0$ and $x^2 - mx + n = 0$ have integer roots, then $n$ is divisible by ?

  1. $7$
  2. $6$
  3. $9$
  4. $5$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For the equations to have integer roots, the discriminants must be perfect squares. Let D1 = m^2 + 4n = k1^2 and D2 = m^2 - 4n = k2^2. Adding these gives 2m^2 = k1^2 + k2^2, and subtracting gives 8n = k1^2 - k2^2. Testing small integer values for m and n shows that n must be a multiple of 6.

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