Multiple choice

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

  1. 6 real roots

  2. at least two real roots

  3. 2 real and 4 imaginary roots

  4. 4 real and 2 imaginary roots
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B Correct answer
AI explanation

The given equation is a product of two quadratic expressions set to zero. According to the zero product property, at least one of the quadratics must equal zero. The discriminant of the second quadratic is s^2 + 8q, which is always greater than or equal to zero for real values of s and q. This guarantees the second quadratic has real roots, meaning the entire equation has at least two real roots.

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