Probability that the equation $x^2 + k_1 x + k_0 = 0$ has real solutions 
$k_1$, $k_0$ are random integer numbers between $1$ and $100$ (including $1$ and $100$, and uniformly distributed). What is the probability that the equation $x^2 + k_1 x + k_0 = 0$ has real solutions?

This is a subproblem of another problem, and I do not know how to approach it without brute force, hope some of you will propose a fresh idea.
The answer should be devised without using computer, if possible.

It has real solutions iff $\Delta=k_1^2-4k_0\ge 0$.
$$P=\frac{\sum_{i=1}^{20}\lfloor\frac{i^2}{4}\rfloor+80\cdot 100}{10000}$$
$(2k+1)^2\equiv 1\pmod{4},\, (2k)^2\equiv 0\pmod{4}$. Therefore (see here and here):
$$P=\frac{\frac{1^2+3^2+5^2+\cdots+19^2-10}{4}+\frac{2^2+4^2+6^2+\cdots+20^2}{4}+8000}{10000}$$
$$=\frac{\frac{10(19)(21)}{3}-10+4\cdot \frac{10(11)(21)}{6}+32000}{40000}=0.8715$$
