Quadratic matrix equation $XAX=B$
Let's assume $A$ is positive definite. Multiplying on left and right by $A^{1/2}$ (the positive definite square root of $A$) the equation becomes $$ (A^{1/2} X A^{1/2})^2 = A^{1/2} B A^{1/2}$$ Now $A^{1/2} B A^{1/2}$ is positive semidefinite, so has a positive semidefinite square root $C$, and we can take $X = A^{-1/2} C A^{-1/2}$ so that $C = A^{1/2} X A^{1/2}$.