Solving a set of two polynomial equations For a given $a,b,c,d$ in $\mathbb{R}$, I want to prove that if $$ac-bd=0 \quad \text{and} \quad ad+bc=0$$ then $a=b=0$ or $c=d=0$. I am able to prove this in a long and cu...--prophetes.ai

Solving a set of two polynomial equations For a given $a,b,c,d$ in $\mathbb{R}$, I want to prove that if $$ac-bd=0 \quad \text{and} \quad ad+bc=0$$ then $a=b=0$ or $c=d=0$. I am able to prove this in a long and cumbersome way, and I'm sure that there is a better way to prove that. Many thanks. Gil.

Multiply $ac=bd$ by $d$ and $ad+bc=0$ by $c$ then

$$bd^2+bc^2=b(c^2+d^2)=0$$

So, either $c=d=0$ or $b=0$. Take the second case then we get $ac=0$ and $ad=0$. So either $a=0$ or $c=d=0$. The second option brings us back to our first case.