Given two quadratic equations (with integer coefficients):
$x^T A_1 x+ b^T_1 x + c_1=0$ and $x^T A_2 x+ b^T_2 x + c_2=0$
The problem is to decide whether they have a common zero. Here $x$ is a vector of real numbers, i.e., we are looking for real solutions.
Q: what is the complexity of this problem?
What we know is that checking whether a family of quadratic equations has a common zero is $\exists R$-complete. However, the current question restricts the family size to be 2. A natural question arises: is this problem simpler, or remains to be $\exists R$-complete?
A related question is here: Zero of a multivariate cubic equation