If x^2+200x+9991=0, which of the following is a value of x?

The crux of solving this question is figuring out how to quickly factorise the equation by splitting the middle term. The numbers are not easy ones such as 5x or 4x.

But there is enough clue in the numbers.

We have to split 200 into two parts, say p and q. p + q = 200 and pq = 9991
The product pq will be maximum when p = q = 100. At that point pq = 10,000 which is more than 9991.
If we add a k to 100 to get p and subtract a k from 100 to get q, the sum will remain 200.

If we do that, we have to find (100 + k)(100 - k) = 9991.
i.e., 10,000 - k^2 = 9991 or k = 3.
So, the value of p and q are 103 and 97. Or the middle term is split as 103 and 97.

Now that we know that 200 splits as 103 and 97, the answer is (x + 103)(x + 97) = 0
So, values of x are -103 and -97