I am given numbers $\beta \in R$, $\alpha_u \in \{-1, 1\}$ and vectors $\mathbf{a}_u$, $u=1,2,\ldots,m$. I have the following relation: $$ \sum_{u=1}^{m}\alpha_u\langle\mathbf{a}_u, \mathbf{x}\rangle^2 = \beta $$ where $\langle\cdot,\cdot\rangle$ is inner product. How can I solve it for $\mathbf{x}$? Are some additional assumptions about the inner-product space which contains those vectors needed in order for the equation to be solvable analytically?
Edit 1: $\mathbf{x}$ is known to be a unit vector, i.e. $\langle \mathbf{x},\mathbf{x} \rangle = 1$.
You have two equations: $$\sum_{u=1}^m\alpha_u\langle\mathbf a_u, \mathbf x\rangle^2 = \beta$$ and $$\langle\mathbf x, \mathbf x\rangle = 1$$
Each equation removes a single degree of freedom in $x$, so if your vector space is $n$-dimensional, the set of solutions is going to be a hypersurface of dimension $n - 2$.
If you are looking for discrete solutions, this will only happen if your space is 2-dimensional. In 3 dimensions, the set of solutions will be a curve. In 4 dimensions it will be a surface.
If your vector space is 1-dimensional, then solutions will usually not exist, unless the $\alpha_u$ and $\mathbf a_u$ happen to be just right.
To find the solutions, convert the problem to one about actual numbers by finding an orthonormal basis for your vector space and expressing the $\mathbf a_u$ in terms of it. Then the two equations become quadratics in the coordinates of $\mathbf x$. Solve for two of the coordinates in terms of the rest. The two equations define the solution hypersurface.