SCHWEITZER 2005 Problem 7

Let $t\in R$ . Prove that $\exists A:R \times R \to R$ such that A is a symmetric, biadditive, nonzero function and $A(tx,x)=0 \,\forall x\in R$ iff t is transcendental or (t is algebraic and t,-t are conjugates over $\mathbb{Q}$ ).