Question

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

Answer 1

We have

$\left(a^2+b^2\right)x^2+2\left(ac+bd\right)x+c^2+d^2=0$

$D=4{\left(ac+bd\right)}^2-4\left(a^2+b^2\right)\left(c^2+d^2\right)$

$=4\left(a^2{c}^2+b^2{d}^2+2acbd\right)-4\left(a^2{c}^2+b^2{d}^2+a^2{d}^2+b^2{c}^2\right)$

$=-4\left(a^2{d}^2+b^2{c}^2-2acbd\right)$

$=-4{\left(ad-bc\right)}^2$

$ifad\ne bc$

$D<0$

$\Rightarrow$ it has no real roots.