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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