[Grue3]

Quadratic equations are common. Very common. In physics, geometry, differential equations and so on. It is also the next step beyond linear equations. It is nice to be able to solve these quickly and to be able to tell their properties just by looking at the coefficients. The sum of roots, the product of roots. The axis of symmetry of the parabola. Better yet, any polynomial of higher degree can be theoretically factored into a product of linear and quadratic polynomials, so it basically always comes down to linears and quadratics.

[eli_gottlieb]

>Better yet, any polynomial of higher degree can be theoretically factored into a product of linear and quadratic polynomials, so it basically always comes down to linears and quadratics.

Could someone have told us that in high school?

[eximius]

Yes, but this is actually false, so it's probably for the best.

Anything of degree 5 or higher is not guaranteed to have solutions solved by radicals (that is, a solution that can be expressed as some rational number to some exponent). For example, x^5 - x + 1 = 0 cannot factored into linear and quadratic polynomials in this way.* The proof for the insolvability is actually quite elegant.

Even so, factoring a polynomial from degree 3 or 4 into quadratics or linear terms is hard. The most general way I can think of is using the rational root theorem and plugging a few values in.

* - You can factor using ultraradicals (yes, it's a thing), but that is far above highschoolers or undergrads, even.

[Grue3]

Just because the solutions are not guaranteed to have a closed form in radicals doesn't mean that my statement is false. In the field of real numbers any polynomial with a degree higher than two is reducible.

http://en.wikipedia.org/wiki/Irreducible_polynomial#Real_and...

[omegaham]

I remember reading (with horror) the methods for factoring cubics and quartics and thinking, "Well, I guess that's what they did before they had Newton's method and computers." Ewwwww.