Question

If the equation $x^2+nx+n=0,nϵI$, has integral roots then $n^2-4n$ can assume

• A. no integral value
• B. one integral value
• C. two integral value
• D. three integral value

Answer 1

Answer: (C)

two integral value

$x^2+nx+n=0,nϵ1$ Discriminant $=n^2-4n$ For integral roots, $n^2-4n$ must be a perfect square.This happens when $n^2-4n=p^2$ for some $pϵI$ here if $p=0$ then $(n-4)n=0\Rightarrow n=4$ or $n=0$ ($2$ integral values)
$n^2-4n-p^2=0$ to have integral solution$\Rightarrow$Discriminant $={(-4)}^2+4p^2$ must be a perfect square$=4(p^2+1)$ must be a perfect square, happens only when $p=0$