Checking whether two quadratic equations have a common zero

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

• I'm guessing from the context that the $x$ is meant to be a vector of reals? – Emil Jeřábek Mar 31 '16 at 8:26
• $x$ is a vector otherwise the problem would be a bit trivial. – maomao Mar 31 '16 at 11:21
• Is x a vector of reals? ​ (as opposed to, say, rationals or complex numbers) ​ ​ ​ ​ – user6973 Mar 31 '16 at 12:10
• it is a convention that we are looking for real solutions. – maomao Mar 31 '16 at 13:00
• Are you aware of any proof that your problem is NP-hard? ​ ​ – user6973 Mar 31 '16 at 21:28