Equations in Self-Similar Groups

We develop a toolkit to decide some quadratic equations with constants in certain self-similar groups and determine conditions under which these tools work. It turns out that for explicitly given groups the question of solvability of some equations reduces to a large but finite number of calculations. For special cases we provide the algorithms in the GAP language and use the computed results to verify that for example the Gupta-Sidki has commutator width at most 3.