Choose the correct answer from the given four options in the following questions: The zeroes of the quadratic polynomial $ x^{2}+k x+k, k 0 $,(A) cannot both be positive(B) cannot both be negative(C) are always unequal(D) are always equal

Given:

Quadratic polynomial \( x^{2}+k x+k, k ≠ 0 \).

To do:

We have to find the nature of the zeroes.

Solution:

Let $p(x)=x^{2}+k x+k$

Here,

Product of zeroes $=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$

$=\frac{k}{1}$

$=k$

The sign is positive it means both the zeroes should have the same sign(both positive or both negative).

Sum of zeroes $=-\frac{\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$

$=-\frac{k}{1}$

$=-k$

The sign is negative and both have the same sign.

This implies, the zeroes are both negative.

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