Can the quadratic polynomial x^{2} + kx + k have equal zeroes for some odd integer k > 1?

No, the quadratic polynomial x^{2} + 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

b^{2} - 4ac = 0

b^{2} = 4ac

k^{2} = 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 x^{2} + kx + k cannot have equal zeroes for some odd integer k > 1

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