Compressing the cubic.

Hi everyone. Still stuck in the hospital, can't got home. Apparently i'm still on antibiotics. (Not the kind that you swallow with water, the kind that goes directly into your veins.) Pretty much nothing to do here except maths and wahjong.. Thanks for the great site lee yen XD. I suck at wahjong tho, 4 wins, 17 losses?!?!?

Anyway i promised in the last blog that i would show how every cubic equation can be reduced to the form x³+px+r

we start with the general cubic x³+tx²+wx+z=0 (if the coefficient of x³ is not one we can just divide through to get one)
then we make the substitution x=y-t/3

(y-t/3)³+t(y-t/3)²+w(y-t/3)+z=0
y³-t³/27-y²t+t²y/3+ty²-2yt²/3+t³/9+wy-tw/3+z=0
see that the y²'s cancel out beautifully :D
y³+(t²/3-2t²/3+w)y+(-t³/27+t³/9-tw/3+z)

And there we have it! The cubic has been reduced to the form y³+py+r=0
with p=(t²/3-2t²/3+w) and r=(-t³/27+t³/9-tw/3+z)

I know it looks devilish, but basically all the technques used are from elementary algbra. In fact this particular solution was found in the early 17 century and obviously the techniques used were already known to the greeks over two thousand years ago. Actually it not all that intimidating if you do it with real numbers instead of t's,w's and z's. (really! i've done it countless of times!)

P.S. kk food sucks, my dad has been sneaking in laksa and mee rubus from downstairs much to the disapproval of the doctors. "Your immune is very low, outside food dunno whether clean or not, easy to get infected, then you'll become very sick"... heck care...