Show that zeroes of polynomial of type x square-5x+k can never be equal in magnitude and opposite in sign

Answer :

We have polynomial :  x ² - 5 x  +  k

And we know quadratic formula :

x  = $\frac{-\mathrm{b} \pm \sqrt{{\mathrm{b}}^2 - 4\mathrm{ac}}}{2\mathrm{a}}$ , Here a = 1 , b  = - 5  and c  = k  , So

x  = $\frac{-\left( - 5\right) \pm \sqrt{{\left( - 5\right)}^2 - 4 \times 1 \times k}}{2 \times 1} = \frac{5 \pm \sqrt{25-4k}}{2}$

So,

Zeroes of given polynomial :

x  = $\frac{5 +\sqrt{25-4k}}{2}$

And

x  = $\frac{5 -\sqrt{25-4k}}{2}$

Here we can easily see that both zeroes will never be equal in magnitude and opposite in sign .