What is the quadratic equation and what does it do?
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A quadratic equation is just a specific type of polynomial equation; one that looks like ax^2 + bx + c = 0, if solving for particular values (called roots). In general, a quadratic just refers to the expression on the left hand side of the equation.
Quadratics are useful for many things. I’d say the two “common” ways they come up in applied situations is, firstly, in problems involving the area of something with unknown side lengths that are related to each other, and secondly in problems involving something analogous to constant acceleration (often it literally is constant acceleration, such as gravity to a good approximation).
In areas outside of applied mathematics, they are everywhere for many many other reasons. In discrete situations, you have the idea of quadratic residues and quadratic non-residues, which for example come up in searching for prime numbers. This furthermore relates to ways that primes and number theory is used in cryptography.
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