Tikhonov Regularization Alternate Formulation
Just expand the term to be minimized, take the derivative with respect to $x$ and set it equal to zero:

$$\begin{align}||Ax-b||_2^2+\lambda^2||x||_2^2&amp;=(Ax-b)^T(Ax-b)+\lambda^2x^Tx\\&amp;=x^TA^TAx-2b^TAx+b^Tb+\lambda^2x^Tx\tag{1}\end{align}$$

Taking the derivative w.r.t. $x$ and setting it equal to zero gives

$$2A^TAx-2A^Tb+2\lambda^2x=0\tag{2}$$

from which the given equation follows.