Question

Prove that the equation $x^2(a^2+b^2)+2(ac+bd)+(c^2+d^2)=0$ has no real root, if $ad\ne bc$.

Answer 1

Given : $x^2(a^2+b^2)+2(ac+bd)+(c^2+d^2)=0$

$\therefore$ $a^{\prime }=(a^2+b^2),b^{\prime }=2(ac+bd),c^{\prime }=(c^2+d^2)$

Discriminant is given by,

$D=b^{\prime 2}-4a^{\prime }{c}^{\prime }$

The discriminant of the given equation is given by
$D=4{(ac+bd)}^2-4(a^2+b^2)(c^2+d^2)\Rightarrow D=4[2ac.bd-a^2{d}^2-b^2{c}^2]$

$\Rightarrow D=-4[a^2{d}^2+b^2{c}^2-2ad.bc]=-4{(ad-bc)}^2$

We have $ad\ne bc$

$\therefore ad-bc\ne 0\Rightarrow {(ad-bc)}^2>0\Rightarrow -4{(ad-bc)}^2<0\Rightarrow D<0$

Hence the given equation has no real roots.