Bound from below with Triangle inequality

Question

This might be a fairly easy question but I am a bit stuck. I am trying to bound from below the following quantity \begin{equation} |A x \cdot x|^{\frac{1}{2}}, \end{equation} where $A$ is a complex-symmetric, $n \times n$ matrix and $x$ is a real $n \times 1$ vector. $\cdot$ is define as the real dot product. Now I have the following ellipticity bounds \begin{align} C_1 |x|^2 \leq A_R x \cdot x \leq C_2|x|^2 \\ C_3 |x|^2 \leq A_I x \cdot x \leq C_4|x|^2 \end{align} where $A_R$ and $A_I$ are the real and imaginary parts of $A$ - that is $A = A_R + iA_I$ and $C_i$ are positive constants. I had the idea to do the following \begin{align} |A x \cdot x|^{1/2} &= |(A_R + iA_I) x \cdot x|^{1/2} \\ &=|A_R x \cdot x + iA_Ix \cdot x|^{1/2} \end{align} This is where I get stuck. Is there a way bound the right-hand side from below, maybe using some form of the triangle inequality for example? I would like something like \begin{align} |A_R x \cdot x + iA_Ix \cdot x|^{1/2} &\geq |A_R x \cdot x|^{1/2} + |i||A_I x \cdot x|^{1/2} \\ &=|A_R x \cdot x|^{1/2} + |A_I x \cdot x|^{1/2} \end{align} Of course this isn't correct but is the any modifications I can make to make it correct.