1. "For real roots, b^2-4ac>0b2−4ac>0:"
A basic quadratic has the formula ax^2+bx+cax2+bx+c. When you solve by completing the square (which is just the rearranged quadratic formula) you end up square rooting b^2-4acb2−4ac. (the discriminant).
If Delta (the discriminant) is less than zero, you are rooting a negative number (which is technically impossible) so you get a complex number and use i.
If the discriminant equals zero, then it disappears (root 0 = 0) and you only have one real root (there could be imaginary roots)
If Delta is bigger than zero, you square root a positive number, and end up with two real roots (because it's ±sqrt).
**2. How this relates to your question, and k(k-4)>0 **
In your equation, b=c; both are the same, k. So the discriminant is k^2-(4*1*k) = k^2-4k = k(k-4), which must be bigger than zero to have two positive real roots as explained above.
3. the union stuff
This is basically a short way of writing what's stated on the 2nd and 3rd => symbols in your image.
(-"infinity",0): k can be from negative infinity to zero. (4,"infinity"): k can also be anywhere bigger than 4.
uu means that these sets are disjoint, there's a gap between them. I think. Someone should double check this bit, I think it's right but I've never really learnt union notation.
