Multiple choice

If the roots of the quadratic equation $ax^2 + bx + c = 0$ are imaginary, then for all values of $a, b, c$ and $x \in R$, the expression $a^2x^2 + abx + ac$ is

  1. positive

  2. non-negative

  3. negative

  4. may be positive, zero and negative
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A Correct answer
Explanation

If roots are imaginary, the discriminant b^2 - 4ac < 0. The expression a^2x^2 + abx + ac can be rewritten by completing the square or analyzing the discriminant of the quadratic in x, which is (ab)^2 - 4(a^2)(ac) = a^2(b^2 - 4ac). Since b^2 - 4ac < 0 and a^2 > 0, the discriminant is negative, meaning the expression maintains the same sign as a^2, which is positive.

AI explanation

Since the roots of ax^2 + bx + c = 0 are imaginary, its discriminant is negative, meaning b^2 - 4ac < 0. The given expression is factored by taking a common term: a^2x^2 + abx + ac = a(ax^2 + bx + c). Because a is the leading coefficient of a quadratic with imaginary roots, a and c must have the same sign. Multiplying the positive quantity a by the expression (ax^2 + bx + c), which always shares the same sign as a, results in a value that is always positive.

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