Minimum Value Of Sum Of Roots

Relevant wiki: Vieta's Formula Problem Solving - Intermediate

Let the two integer roots be $m, n$
$mn = 2016 , m+n = a$
We know that,
$(m+n) = \sqrt{4mn + (m-n)^{2}}$

$a = \sqrt{8064 + (m-n)^{2}}$
To minimize $a$ we have to minimize $|m-n|$ . The two closest factors of 2016 are $42, 48$
$(m,n) = (42,48) ,(48,42)$
$a_{min} =\sqrt{8064+36} = 90$

Suppose the two positive integers roots are $p$ and $q$ . Then $pq=2016$ and $a=p+q$ .

Since $\sqrt{2016} \approx 44.9$ , then $a = p+q \ge 2\sqrt{pq} =2\sqrt{2016} > 89.8$ . As $p$ and $q$ are integers, $a \ge 90$ , which is possible if $p=42$ and $q=48$ .

Now, let the positive integer solutions be $p$ and $q$ .

We know that $p+q=a$ and $pq=2016$ .

This means that we are looking for a pair of factors of $2016$ which has the smallest sum.

Notice that for the number $30$ :

$1 \times 30 = 30\\ 2 \times 15 = 30\\ 3 \times 10 = 30\\ 5 \times 6 = 30\\ 6 \times 5 = 30\\ 10 \times 3 = 30\\ 15 \times 2 = 30\\ 30 \times 1 = 30$

We can see that once we go past $\sqrt{30} \approx 5.48$ , the factors will repeat themselves. Also, notice that the factors closest to $\sqrt{30}$ (in this case, the pair of factors $5$ and $6$ ) will give the smallest sum of factors. You can try this with other numbers, the results will be the same.

This implies that we are looking for the pair of factors closest to $\sqrt{2016} \approx 44.9$

Test a few values from $44$ onwards, and you will find that the closest pair of factors to $\sqrt{2016}$ is $42$ and $48$

Therefore, $a=42+48 = \boxed{90}$

Since the equation has 2 integer solutions
$D>0 \implies a^2 - 4 \times 2016$ should be perfect square, minimum value satisfying this is 36.
Therefore $a^2 = 4 \times 2016 + 36 = 8100$ . Therefore $a = 90$