College Algebra/Solving quadratic equations

A quadratic equation is any equation of the form $$ ax^2 + bx + c = 0 \,\;$$ representing a parabola in the plane. Solving quadratic equations and finding their zeros is a very important ability in mathematics, allowing mathematicians to work with equations involving larger exponents effectively. There are several methods for solving quadratic equations which are all simple to use but vary somewhat in speed and the sets of numbers for which the method is effective.

Basic principles
A equation is only quadratic if it can be put into the form $$ ax^2 + bx + c = 0 \,\;$$ where $$ a \,\;$$ is not equal to 0. This means that the equation $$x^2 - 5 = 2x \,\;$$ is quadratic because it can be put into the form $$ x^2 - 2x - 5 = 0 \,\;$$, but the equation $$ x^3 - 6x - 5 = 0 \,\;$$ is not because it contains a cube and so cannot be put into a quadratic form.