quadratic in disguise (idea)

I love the quadratic equation, don't you? If ax² + bx + c = 0, then:

        -b + root( b2 - 4ac )
 x = -----------------------
               2 a

How aesthetic. Well, anyway, I'm here to talk to you about quadratics in disguise! These little buggers can come up often when you're trying to solve something, but you just don't notice! So, whip out your TI-83 (with installed Quadratic equation program) and look at these examples:

(a^4)/2 + a^2 + 1 = 0
You can rewrite this as u^2 /2 + u + 1 = 0, then solve for u and take the square root of those answers. (Don't forget that a^2 = u has two answers for every u.

5cos²(x) + sin(x) - 5 = 0
Which can be simplified:
-5(1 - cos²(x)) + sin(x) = 0
-5sin²(x) + sin(x) = 0
-5u² + u = 0
Which you can solve. This actually has some complicated answers, due to the sin(x).

5x + 4 / x = 12
Multiply by x and subtract 12x on both sides:
5x² - 12x + 4 = 0
Which is mere child's play.

So, the quadratic formula shows up in all sorts of nifty (and not so nifty) places. Keep a look out for it, because it might save your life!