Can the quadratic polynomial x² + kx + k have equal zeroes for some odd integer k > 1?

No, the quadratic polynomial x² + kx + k cannot have equal zeroes for some odd integer k > 1

Let suppose, if it has equal zeroes

the zeroes of a quadratic polynomial are equal when discriminate is equal to 0

i.e., D = 0

b² - 4ac = 0

b² = 4ac

k² = 4k

k = 0 or k = 4

But it is given that k > 1, so we reject the value k = 0 < 1

∴ k = 4

But 4 is not an odd number and only at k = 4 will the polynomial get equal roots.

Hence, the quadratic polynomial x² + kx + k cannot have equal zeroes for some odd integer k > 1