Why we need to know how to solve a quadratic? 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. 
 A: Others have echoed this sentiment, but I'll add some emphasis on non-mathematical applications. Elementary algebra is really the first place where students (should) learn how to solve problems, rather than simply answer questions.
By learning how to solve quadratics -- perhaps not through direct, rote application of the quadratic equation --  one learns how to solve problems. By learning about the properties of polynomial equations, one learns how to solve problems. By learning about the behavior of trigonometric functions, one learns how to solve problems.
These are tools useful for math, for sure, but more importantly, they teach us to look at details, learn about the importance of essential properties, and how to formulate solutions. These are skills used in day-to-day life.
The math might never be used, but the skills will be. The skill of how to approach a problem from different possible directions and formulate a solution strategy is useful in many quotidian tasks.
For instance, I recently moved. Moving has not a whole lot to do with math. But, the task of packing my things, moving them across town into a new place, and integrating those things into a slightly different lifestyle poses a problem. Sure, I could have just put all my things into boxes, shoved it into a truck, and thrown it into the house. But probably that would have been a whole lot more work than looking at the essential properties of my belongings (some are fragile, some are large, some belong upstairs, some go in the basement), and formulating a scheme that incorporates these essential properties into a comprehensive solution strategy. And in the end, it was a lot less work than it might have been.
Algebra teaches one how to evaluate a problem, formulate a strategy based on some fixed rules (e.g. sofas are heavy), and implement that strategy to a verifiable result.
As a topic for another forum, I feel that the problem is not so much with math education (though that takes a share of the blame) but with pre-algebra education: by the time students reach algebra, they've undertaken 6 or 7 years of education where all they need to do is deliver answers, not strategies.
So, to the original question: why know how to solve a quadratic? Because learning how to solve a quadratic gives you the skills to solve myriad problems from myriad domains of quotidian life by leveraging known rules and essential properties.
A: While it may not answer your question, this is a perennial problem that reared its head recently in the New York Times. Among lots of responses, this one is worth reading.
A: You need to know how to do math at a decent level to perform in any science or technology job.  But I'll be more specific: in chemistry, you explicitly need quadratic equations to solve acid/base equilibrium problems: i.e. the degree to which an acid or base dissociates in water.  Even more specifically, you need acid/base equilibrium math to do calculations for buffered solutions: i.e. the amount of a conjugate base needed in combination with its acid to maintain pH in a narrow range, which is extremely important in biological systems. Sure, you say calculators can solve these problems, but you need to understand quadratic equations to set up the problem in the first place.
A: I recommend reading the Plus Magazine article about quadratic equations:
http://plus.maths.org/content/101-uses-quadratic-equation . It sheds some historical perspective on the quadratic equation, including some problems that could have been the motivation for solving it - for example, how do you size a field in order to obtain a certain amount of crop production? 
Also, the quadratic equation is often found in all kinds of day-to-day applications - how high do I need to aim a water jet for it to reach a certain landing spot? How much fence do I need to buy for a triangle shaped field? Or who could forget the parabolic antenna? 
Answering the more general question, I think learning about history is the best way to have these kinds of answers. Almost every mathematical construct taught at college level was either devised to solve a practical problem, found applicability in various domains once it was found, or at the very least has some interesting backstory preceding or following its discovery. All of these help give some concrete context to the abstract mathematics.
Of course have in mind that being able to answer these questions will probably not solve your students' interest issues by itself!
A: I'll try to answer your question from another point of view. To me, it's important not to know how to solve quadratic equations, but to show your students that they already know how to do that. 
Consider the following rectangle as an easy way to show that $(x+y)(a+b+c+d)$ is indeed a sum of smaller rectangles formed by those lines and segments:

(source: popmath.ru) 
The same picture show that any expressions of that sort will consist of $2 \cdot 4$ members, which can come quite handy while exlaining the intuition behind binomial formula. 
Did your student have to know anything to come up with this conclusion? No, he already knew that.
Going on, the same can be said about quadratics, consider that $(a+b)^2$ expression is beautifully summed up in this picture:

(source: popmath.ru) 
So to come up with a way of solving it your student only needs to understand that any equation with $a^2$ is a crippled square of sum formula, thus by completing the square he can solve it (at least in trivial case without any fancy numbers) by not memorizing anything.
And that's the point of math — it can be treated as an art of arriving at unexpected conclusions by reflecting on the things you already know. Quadratics just serve as a simple example of that approach to make sure your students get the general idea.
Hope that helped to answer your question.
A: 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).
A: 
The Universe is a grand book which cannot be read until one first  learns to comprehend the language and become familiar with the 
  characters in which it is composed. It is written in the language  of
  mathematics…

(attributed to Galileo)
Sounds like enough encouragement for the receptive.
A: I remember this exact question from every math class I've ever been in.  After having kids and thinking more about why the question was asked I found that the best answer is this:  Mathematics is a method for explaining and finding solutions to not only problems, but the very foundation of everything that exists.  No your students will probably never have to use the quadratic formulae unless they become chemists or some such field, however understanding that the formulas they learn apply to everyday life and the world around them will help them better understand the whys and hows of reality.  There is no other way that is not conjecture.  Math is fact.  
