You never need the quadratic equation if you know other methods for solving quadratics. You can also try factoring or completing the square (completing the square is similar to using the quadratic equation, since the derivation of the quadratic equation involves completing the square of a general quadratic). However, in general, an equation is called an implicit definition for . When you write , you do not give an explicit definition for . In order to have an explicit definition (to find the actual value for a variable), you need that variable on one side of the equation and the other side to be a function that does not include the variable you solved for. So, with Plato's and Soroban's solution, they have on one side of the equation and a function that does NOT involve on the other (making their solution an explicit definition for , allowing them to find an actual value for it).
You don't need to use the quadratic formula or even write it as a quadratic equation (if that was your question).
We can write and
So
so that
Of course, that isn't one of the forms given but you can write in terms of a logarithm, basically by solving a quadratic equation as you have been told!