I need help, my teacher says that I'm half way right with this but not fully what am I missing I'm confused.

The criteria for the quadratic equation as i written is:

y>0 is real roots
y=0 is real and equal roots
y<0 is imaginary or complex roots

My teacher said I'm half right, but still missing something what am I missing?

for any quadratic in the form

ax^2 + bx + c = 0
the discriminant, which is b^2 -4ac, determines the nature of the roots

if you replace y with either "the discrimiant" or with b^ - 4ac, then your teacher will probably be happy with your answer.

It is likely that your teacher did not like the use of the variable y in your answer.