I love the

quadratic equation, don't you? If ax

^{2} + bx + c = 0, then:

-b __+__ root( b^{2} - 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.

5cos

^{2}(x) + sin(x) - 5 = 0

Which can be

simplified:

-5(1 - cos

^{2}(x)) + sin(x) = 0

-5sin

^{2}(x) + sin(x) = 0

-5u

^{2} + 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:

5x

^{2} - 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!*