I need to solve the following equation to find $x$ in $\mathbf{x}^T=[1,-x]$
$$(E[\mathbf{x^TAZ}])^2= \alpha^2Var[\mathbf{x^TAZ}]$$
where $\mathbf{Z} \sim N(\mathbf{m},\mathbf{\Sigma})$ is a vector of independent normal random variables, i.e $\mathbf{\Sigma}$ is a diagonal matrix.
One way of doing this is to express everything as scalars and to solve a quadratic qaution, but is there a way to do it with vector algebra? Maybe using a Moore-Penrose inverse? I only got to
$$(\mathbf{x^TA}E[\mathbf{Z}])^2= \alpha^2\mathbf{x^TA \Sigma A^Tx}$$
or by setting $\mathbf{x^TA} = \mathbf{c^T}$
$$(\mathbf{c}^T \mathbf{m})^2= \alpha^2 \mathbf{c}^T \mathbf{\Sigma c}$$
and not sure how to proceed from here (if it is even possible to do so).