Answer:
n < -15/4
Explanation:
You want to use the discriminant to find the values of n for which the quadratic 3z² -9z = (n -3) has only complex solutions.
Discriminant
The discriminant of quadratic equation ax²+bx+c = 0 is ...
d = b² -4ac
The given quadratic can be put in this form by subtracting (n-3):
3z² -9z -(n -3) = 0
This gives us ...
and the discriminant is ...
d = (-9)² -4(3)(-(n-3)) = 81 +12(n -3)
d = 12n +45
Complex solutions
The equation will have only complex solutions when the discriminant is negative:
d < 0
12n +45 < 0 . . . . . use the value of the discriminant
n +45/12 < 0 . . . . . divide by 12
n < -15/4 . . . . . . . subtract 15/4
There will be two complex solutions when n < -15/4.