Why we need to know how to solve a quadratic?
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).
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.
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.