Multiple choice

The quadratic equation $\displaystyle a x^{2}+b x +c=0$ will always have imaginary root in which of the following cases?

  1. $a < -1 , \ \ 0 < c < 1, \ \ b > 0$
  2. $a < -1, \ \ 1 < c < 0, \ \ 0 < b <1$
  3. $a < -1 , \ \ c < 0, \ \ b > 1$
  4. $a < -1, \ \ c < - 1, \ \ 1 < b < 2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A quadratic has imaginary roots when its discriminant is negative, so b^2 - 4ac < 0. In option D, ac > 1 and b^2 < 4, making 4ac > b^2 and the discriminant negative.

AI explanation

A quadratic equation has imaginary roots when its discriminant, b^2 - 4ac, is negative. Testing option D with a less than -1, c less than -1, and b strictly between 1 and 2 gives a positive value for the product ac, making the entire term -4ac negative. Since b is less than 2, b^2 must be less than 4, and b^2 minus a larger positive number will always result in a negative discriminant. This condition guarantees the equation will always have imaginary roots.

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