Five years ago I was tutoring orphans in a local hospital. One of them asked me the following question when I tried to ask him to solve a quadratic:
Why do I need how to solve a quadratic? I am not going to use it for
my future job!
This question is, largely not mathematical. Substituting 'quadratic' with 'linear forms' or 'calculus' or 'Hamlet' would not make much difference since the specific knowledge is not used on a day to day basis in most occupations except academia. But I feel puzzled as how to justify myself that 'learning quadratics is important enough that you must learn it'. At that day, I used a pragmatic argument that he need to pass various qualification exams to get to college, and after college he can find a job he wants. But this feels self-defeating – we are not learning for the sake of passing tests or getting high grades. I do not know how to make the kid understand that "knowing how to solve a quadratic is interesting and knowing how to solve higher degree ones can be awesome" – because knowing $(x-p)(x-q)=x^2-(p+q)x+pq$ is not very interesting to him.
Since I am still puzzled over it I decided to ask others who may had similar experience. What do you say when others ask you "what is the benefit of knowing $xxx$ theorem? Will you respond that "knowing $xxx$ is helpful/interesting because of $a,b,c,d$ reasons?"(thus refute the utilitarian argument), or arguing as this post that some knowledge is essential to know for anyone?
My father asked me "What is the importance of proving $1+1$ (the Goldbach Conjecture)" when I returned from college. I do not know how to answer as well even though I know the history behind the conjecture. Now I am going to become a teaching assistant, I think I should be able to answer such questions before I am at the stage and someone ask me questions like "Why do I need to know calculus"? again. So I post this at here.
Best Answer
I don't think solve a quadratic equation is extremely important in and of itself.
What is rather important, however, is the abstract skill of recognize a problem as an instance of a problem type for which you've heard of a canned solution, and apply the canned solution formula by plugging in parameters from the particular problem instance. Many more people will need that than will need the specific skill of solving quadratics.
The quadratic formula is a nice elementary example of a problem type that is usually easier to solve by plugging into a formula than by remembering a derivation. It is fairly clear whether a problem is an instance of the one it solves, for example, so doesn't need a long touchy-feely discussion about whether or not it is reasonable to solve this or that problem as a quadratic equation in the first place. (Such deliberations also need to be taught and learned, of course, but preferably after the mere art of plugging-into-formulas has become a trivial skill).