Multiple choice

Let $f(x)$ be a quadratic function given by the equation$f(x)=a{x^2}+bx+c$. The value of the function at point $x=-\dfrac{b}{2a}$ is less than zero. If $a>0$, the nature of roots of the given equation are:

  1. Real and Equal

  2. Imaginary

  3. Real and Distinct

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The vertex of the parabola f(x) = ax^2 + bx + c is at x = -b/2a. The value of the function at the vertex is the minimum value (since a > 0). If the minimum value is less than zero, the parabola must cross the x-axis at two distinct points, meaning the roots are real and distinct.

AI explanation

The x-coordinate of the vertex is -b/2a, and evaluating the function there gives a minimum value less than zero. Since a > 0, the parabola opens upwards, meaning the vertex lies below the x-axis. A parabola opening upwards with its lowest point below the x-axis must cross the x-axis twice, guaranteeing two real and distinct roots.

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