Multiple choice

If $p,q,r,s \in R,$ then the equation $(x^{2}+px -3q)(x^{2}-rx+q)(x^{2}-sx+2q)=0$ has

  1. exactly six  real roots

  2. at least three real roots

  3. at least four real roots

  4. at least two real roots
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D Correct answer
Explanation

Each quadratic factor has a discriminant. For (x^2 + px - 3q), D1 = p^2 + 12q. For (x^2 - rx + q), D2 = r^2 - 4q. For (x^2 - sx + 2q), D3 = s^2 - 8q. If q > 0, at least one discriminant must be non-negative. If q < 0, the first one is positive. Thus, at least two real roots are guaranteed.

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