Final answer:
The quadratic expression ax²+3x−6 will have two imaginary roots for any value of a less than -3/8.
Step-by-step explanation:
To find the values of a that will result in the quadratic expression ax²+3x−6 having two imaginary roots, we can use the discriminant of the quadratic formula. The discriminant is defined as b²-4ac in the quadratic formula. In this case, our formula becomes 3²-4a(-6). For the quadratic expression to have two imaginary roots, the discriminant must be less than zero. So we have 9+24a<0.
Solving this inequality, we find that a<-3/8. Therefore, for any value of a less than -3/8, the quadratic expression will have two imaginary roots.