Multiple choice

The roots of the equation $x^2 + (p + 2)x + 2p = 0$ are distinct integers when-

  1. $ p \ \epsilon \ I$
  2. $ p \ \epsilon \ I$ - {0}
  3. $ p \ \epsilon \ I$ - {2}
  4. $ p \ \epsilon \ I$ - {1}
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The polynomial factors as (x + 2)(x + p), so its roots are -2 and -p. These roots are integers when p is an integer, and they are distinct when p is not 2.

AI explanation

By factoring the quadratic expression by grouping, x^2 + (p + 2)x + 2p becomes x(x + 2) + p(x + 2), leading to (x + p)(x + 2) = 0. This shows the roots of the equation are -p and -2, which are integers if p is an integer. However, if p is exactly 2, the roots become -2 and -2, which are not distinct integers. Therefore, the roots are distinct integers for any integer p except 2.

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