How can I guarantee the existence of a solution to this quadratic system of equations?

I have $$n$$ real quadratic equations and $$n$$ real variables, $$x_i$$, of the following form:

$$\sum_{i\neq j} a_{ijk}x_ix_j+\sum_ib_{ik}x_i+c_k=0 \ \forall k$$

for $$i,j,k\in\{1,\dots n\}$$; all coefficients are real and I am interested in the existence of a real solution. Is there a condition on the coefficients to guaranty this?

• Some jargon: real Nullstellensatz (due to Krivine, Dubois, and Risler), real ideal, real radical ideal, real Nullstellensatz certificate. – Ricardo Buring Jun 17 '19 at 9:07