A discriminant is when in the Quadratic Formula, the part that is b^2-4ac under the standard form of ax^2+bx+c

now if we want the discriminant to be positive b^2-4ac has to be positive, so anytime b^2 is greater than 4ac (because they are being subtracted). So Pick any number, I'll choose 5 for b, and 1 for a and c. Your equation will be x^2+5x+1. The discriminant is positive, 5^2-4(1)(1) = 21.

Likewise for the other two examples, for it to be 0 they must be equal, Lets choose 2 for b and 1 for a and c, 2^2 - 4(1)(1)=0, thus we get the formula x^2+2x+1.

Finally for it to be negative, 4ac must be greater, lets have a b and c equal to 1, meaning 1^2 - 4(1)(1) = -3, giving you an equation of x^2+x+1.