Efficient approach to verifying that the only real solutions to this system of quadratic equations are trivial? Let $a,b,c,d,e,f$ be real numbers. In the course of my research, I've arrived at the following system of equations \begin{eqnarray*}
a+d+f &=& 0, \\
ad + af + df - b^2 - c^2 - e^2 &=& 0 \\
ae^2 + c^2 d - 2bc - adf &=& 0.
\end{eqnarray*}
Inserting this into Wolfram, we get some unwieldy expressions, but all of them require that one of these numbers is complex.

Is there an efficient method for proving directly that the only real solution to this system of equations is the trivial one?


Squaring the first equation you get
$$a^2 + d^2 + f^2 + 2ad + 2af + 2df = 0.$$
Substituting in the second one gives
$$\tfrac12 a^2 + b^2 + c^2 + \tfrac12 d^2 + e^2 + \tfrac12 f^2 = 0,$$
but this is possible only when all the variables are equal to $0$.
