Question

If the roots of x^2 - px + q = 0 are two consecutive integers, prove that p^2 - 4q -1 = 0

Answer 1

sum = p

product = q
let a & b the roots
a + b = p
ab = q
since a & b are consecutive integers (i.e. b = a+1):
2a + 1 = p
a^2 + a = q
substitute.
p^2 - 4q -1 = 0
4a^2 + 4a + 1 - 4a^2 + 4a - 1 = 0
indeed,
0 = 0