determine one value of c that allows the trinomial cy^2 + 36y - 18 to be factored over the integers
Use the quadratic formula:
y=(-36±sqrt(36^2+4*18*c))/2c
For rational solutions,
36^2+4*18*c
=1296+72c
must be a perfect square.
Solve for c by trial and error or as follows:
Perfect squares as of 1296 are:
1296
1369
1444
1521
1600
..
or
1296+73k+k(k-1)
where k=1,2,3,...
So we need to solve for integer values of c in:
1296+72c=1296+73k+k(k-1)
c=(73k+k(k-1))/72
=k+(k^2)/72
for k=12,
c=12+144/72=14
Check:
14y^2 + 36y - 18
=2(y+3)(7y-3)