The zeroes of the quadratic polynomial \(x^2 + kx + k\), where \(k \ne 0\), are:
cannot both be positive
cannot both be negative
are always unequal
are always equal
Step 1: Recall relations between coefficients and zeroes.
For a quadratic \(x^2 + kx + k\):
Sum of zeroes = \(-k\).
Product of zeroes = \(k\).
Step 2: Assume both zeroes are positive.
If both are positive, then:
Sum > 0 ⇒ \(-k > 0\) ⇒ \(k < 0\).
Product > 0 ⇒ \(k > 0\).
This is a contradiction (\(k\) cannot be both positive and negative at the same time).
Step 3: Conclusion.
It is impossible for both zeroes to be positive.
Correct Option: (A) cannot both be positive