How do you convert standard form to vertex form?

How do you convert a quadratic equation from its standard form, ax^2 + bx + c = 0, to its vertex form, a(x-h)^2 + k = 0? I'm struggling to understand the process and I would appreciate a step-by-step explanation to help me grasp the concept better. Are there any tricks or mnemonics that you recommend to make the conversion process easier? Additionally, could you explain how understanding vertex form can be beneficial in solving quadratic equations and analyzing their graphs?