Multiple choice

Consider the quadratic equation $ax^{2} + bx + (a^{2} + b^{2} + c^{2} - ab - bc - ca) = 0$ (where $a,b, c$ are distinct real numbers). If the graph of the quadratic polynomial is mouth opening downwards, then the product of roots of the corresponding quadratic equation is always

  1. positive

  2. negative

  3. depends on the sign of b

  4. can't determine
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B Correct answer
Explanation

For a quadratic ax^2 + bx + c = 0, the product of roots is c/a. Here, c = a^2 + b^2 + c^2 - ab - bc - ca = 1/2 * ((a-b)^2 + (b-c)^2 + (c-a)^2). Since a, b, c are distinct, c is always positive. Given the parabola opens downwards, a must be negative. Thus, the product of roots (c/a) is positive/negative = negative.

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