Multiple choice

If ${z}{1}$ and ${z}{2}$ are roots of quadratic equation $a{z}^{2}+bz+c=0$ such that $Im({z}{1}{z}{2})\ne 0$ then

  1. $a,b,c$ are all real
  2. atleast one of $a,b,c$ is real
  3. atleast one of $a,b,c$ is imaginary
  4. $a,b,c$ are all imaginary
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If the coefficients a, b, c are all real, the roots must be either real or complex conjugates. If they are complex conjugates, their product z1*z2 = c/a is real, so Im(z1*z2) = 0. Since Im(z1*z2) is not 0, the coefficients cannot all be real.

AI explanation

Let the roots z1 and z2 have an imaginary product, meaning Im(z1*z2) does not equal zero. By Vieta's formulas, the product of the roots z1*z2 equals c/a. If the imaginary part of c/a is non-zero, then c/a cannot be a real number. This implies that the ratio c/a is imaginary, which is only possible if at least one of the coefficients a, b, or c is imaginary.

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