MHB What are the solutions if h+k=2016 and the roots of x^2+hx+k=0 are integers?

[Albert1]

$if$ $ h+k=2016$
and all roots of $ x^2+hx+k=0$ are both integers
find its solutions

 

[kaliprasad]

$if$ $ h+k=2016$
and all roots of $ x^2+hx+k=0$ are both integers
find its solutions

Spoiler

let the roots be - a and - b and |a| <= |b| so we get
$(x+a)(x+b) = x^2+hx + k$
or $x^2 + (a+b) x + ab = x^2 + hx + k$
so $a+ b = h,ab = k$
hence $(1+a)(1+b) = (1+ a + b + ab) = 1 + h + k = 2017$
it is a prime so 1 + a = 1 and 1 + b = 2017 so a = 0 and b = 2016 and roots are 0 and - 2016. h = 2016 and k = 0 one solution
or 1 + a = - 1 and 1+b = - 2017 giving a = -2 and b = - 2018 another solution so 2 solutions