Let, both $a$ and $b$ belong to the set {1,2,3,4}. What is the number of equations of the form $ax^2+bx+1=0$ which have real roots. Let, both $a$ and $b$ belong to the set {1,2,3,4}. What is the number of equations of the form $ax^2+bx+1=0$ which have real roots.
for real roots, $a \gt 0$, $b^2-4{a}{c} \ge 0$
Here we have $c=1$, and $a \ge 0$
Now we need to have $b^2-4a \ge 0$
i.e. $(-2\sqrt{a} \ge b) \cup (b \ge 2\sqrt{a})$

There are of course 16 equations of the form $ax^2+bx+1=0$ with $a,b \in X:= \{1,2,3,4\}$. Which of them have real roots? 
As you note, we have to determine for which $a,b\in X$ we have $b^2-4a\ge 0$.
So if $a=1$, we have $b=2,3,4$. 
If $a=2$, we have $b=3,4$, if $a=3,4$ then we must have $b=4$. Summing up there are 7 equations. 
