If both $a$ and $b$ belong to the set $\{1,2,3,4\}$ , then number of equations of the form $ ax^2+bx+1=0$ having real roots is
$a.)\ 10\\ \color{green}{b.)\ 7}\\ c.)\ 6\\ d.)\ 12\\ $
To solve this I had to make a table and check each of the $16$ cases.
$$\begin{array}{|c|c|c|} \hline a & b & b^2-4a\geq 0 \\ \hline 1 & 1 & \\ \hline 1 & 2 & \checkmark \\ \hline 1 & 3 & \checkmark \\ \hline 1 & 4 & \checkmark \\ \hline 2 & 1 & \\ \hline 2 & 2 & \\ \hline 2 & 3 & \checkmark \\ \hline 2 & 4 & \checkmark \\ \hline 3 & 1 & \\ \hline 3 & 2 & \\ \hline 3 & 3 & \\ \hline 3 & 4 & \checkmark \\ \hline 4 & 1 & \\ \hline 4 & 2 & \\ \hline 4 & 3 & \\ \hline 4 & 4 & \checkmark \\ \hline \end{array}$$
But I would like to know if there is any short method for it.
I have studied maths up to $12$th grade, thanks.
For real roots,
$b^2\ge{4a}$
Put $a=1$.
$b^2\ge4$
$b$ can be $\{2,3,4\}$.
Now put $a=2$
$b$ can take two values.
Put $a=3$, $b$ can take one value and finally put $a=4$. Here $b$ can take one value.
Hence there are $7$ ordered pairs $(a,b)$ so there are $7$ such equations