Square of 2-norm

Question

This might be silly but how would I solve the following equation for $x$?

$$ \left\| Y - \frac{Z_i}{x} \right\|^2_2 = 2t $$

Answer 1

Write squared norm as inner product: $$\| Y - Z_i/x\|^2 = \|Y\|^2 - \frac{2}{x} \langle Y, Z_i\rangle +\frac{1}{x^2}\|Z_i\|^2 $$ Equating this to $2t$ yields a quadratic equation for $1/x$. (Or a quadratic equation for $x$, if you multiply everything by $x^2$).