x = 1 is a correct answer to the problem. If you substitute it into the original question, you get 2 = 2, correct.
x = 6 has a little problem. When you substitute it into the original question, you get 3 = -3. Oops.
When you square an equation, you lose the sign information. Both positive and negative roots will come out as answers. But in the original problem we assume positive roots.
Here 6 is called an extraneous root. Your mathematics is right but you missed the vital final step -- you MUST check answers in the original equation.
As far as ab = 0 implies either a = 0 or b = 0. this is a basic property of real numbers which you know well. Consider your multiplication tables and also decimals or fractions. What is the only way you get zero as a product?
Finally your work is good and organized. The editing done above is also extremely important. You must do one step per line, only, and put each line directly below the one you derived it from. That way people can read your work. If it as mashed together nobody can make sense out of it.