Question 8

The zeroes of the quadratic polynomial $x^2+kx+k$ where $k\ne 0$, are : (1)cannot both be positive (2)cannot both be negative (3)are always unequal (4)are always equal

Solution:

Let $p\left(x\right)=x^2+kx+k,k\ne 0$
On comparing $p\left(x\right)$ with $a{x}^2+bx+c$, we get
$a=1,b=k$ and $c=k$
Now, $x=[\frac{-}{b}$ [by quadratic formula]
$=[\frac{-}{k}$
$=[\frac{-}{k},k\ne 0$
Here, we see that
$k(k-4)>0$
$\Rightarrow k\in (-\infty ,0)\cup (4,\infty )$
Now, we know that
In quadratic polynomial $a{x}^2+bx+c$
If $a>0,b>0,c>0$ or $a<0,b<0,c<0$,
then the polynomial has always all negative zeroes. and is $a>0,c<0$ or $a<0,c>0$, then the polynomial has always zeroes of opposite sign.
Case I if $k\in (-\infty ,0)$ i.e., $k<0$
$\Rightarrow a=1>0,b,c=k<0$
So, the zeroes are of opposite sign.
Case II If $k\in (4,\infty )i.e.,k\ge 4$
$\Rightarrow a=1>0,b\ge 4$
So, both zeroes are negative
Hence, in any case zeroes of the given quadratic polynomial cannot both be positive.