stickman lore Official 🍔 | 2048 / 0 to ????? Inf

Linear
ax+b=0
ax=-b
x=-b/a
Quadradic
Ax²+bx+c
x²+(b/a)x+(c/a)=0
Substitute x=y-h
A(y-h)²+b(y-h)+c what h make linear coefficient 0 is b/2a
So we Substitute x=y-(b/2a)
This looks like something
(y-(b/2a))²+(b/a)(y-(b/2a))+(c/a)
y²+(4ac-b²)/(4a²)=0
y²=-(4ac-b²)/(4a²)
y=±sqrt(-(4ac-b²)/(4a²))
y=±sqrt(-(4ac-b²))/2a
y=sqrt((b²-4ac))/2a
x=y-(b/2a)->x+(b/2a)=y
x+b/2a=±sqrt((b²-4ac))/2a
x=(-b±sqrt((b²-4ac)))/2a
Cubics
Cubic formula
ax³+bx²+cx+d=0
x=y-b/3a
Expanding it gives us x³+px+q for p=(c - b²/3a), q=((2b³-9abc+27a²d)/27a²)
x³+px+q=0
x³+px+q=0
(a+b)³=a³+3ab(a+b)+b³
x=u+v
(u+v)³+px+q=0
(a+b)³-3ab(a+b)=a³+b³
(u+v)³+p(u+v)+q=0
(u+v)³+p(u+v)=-q
a=u, b=v
(a+b)³+p(a+b)=-q
a³+b³=-q
3ab=p -> ab=p/3
ab=p/3
a³b³=p³/27
K=a³, L=b³
K+L=-q, KL=p³/27
(-q/2+B)+(-q/2-B)=-q
(-q/2+B)(-q/2-B)=p³/27
-q²/4 - B²=p³/27
B² - q²/4=-p³/27
B²=-p³/27 + q²/4
B=sqrt(-p³/27 + q²/4)
K=-q/2+sqrt(-p³/27 + q²/4)
L=-q/2-sqrt(-p³/27 + q²/4)
Quadradic formula form
K=(-q+sqrt(-4p³/27 + q²))/2
L=(-q-sqrt(-4p³/27 + q²))/2
And
a³=(-q+sqrt(-4p³/27 + q²))/2
b³=(-q-sqrt(-4p³/27 + q²))/2
For
a=cbrt((-q+sqrt(-4p³/27 + q²))/2)
b=cbrt((-q-sqrt(-4p³/27 + q²))/2)
It thr
u=cbrt((-q+sqrt(-4p³/27 + q²))/2)
v=cbrt((-q-sqrt(-4p³/27 + q²))/2)
frst root of depressed
x=cbrt((-q+sqrt(-4p³/27 + q²))/2)+cbrt((-q-sqrt(-4p³/27 + q²))/2)
Ratonial root therom
Is x²+cbrt((-q+sqrt(-4p³/27 + q²))/2)+cbrt((-q-sqrt(-4p³/27 + q²))/2)x+cbrt((-q+sqrt(-4p³/27 + q²))/2)+cbrt((-q-sqrt(-4p³/27 + q²))/2)=0
To (-(cbrt((-q+sqrt(-4p³/27 + q²))/2)+cbrt((-q-sqrt(-4p³/27 + q²))/2))±sqrt((cbrt((-q+sqrt(-4p³/27 + q²))/2)+cbrt((-q-sqrt(-4p³/27 + q²))/2))²-4(cbrt((-q+sqrt(-4p³/27 + q²))/2)+cbrt((-q-sqrt(-4p³/27 + q²))/2))))/2
And
Final!!!
cbrt((-q+sqrt(-4p³/27 + q²))/2)+cbrt((-q-sqrt(-4p³/27 + q²))/2)+b/3a
((-(cbrt((-q+sqrt(-4p³/27 + q²))/2)+cbrt((-q-sqrt(-4p³/27 + q²))/2))±sqrt((cbrt((-q+sqrt(-4p³/27 + q²))/2)+cbrt((-q-sqrt(-4p³/27 + q²))/2))²-4(cbrt((-q+sqrt(-4p³/27 + q²))/2)+cbrt((-q-sqrt(-4p³/27 + q²))/2))))/2)+b/3a
Into the!!!
cbrt((-((2b³-9abc+27a²d)/27a²)+sqrt(-4(c - b²/3a)³/27 + q²))/2)+cbrt((-((2b³-9abc+27a²d)/27a²)-sqrt(-4(c - b²/3a)³/27 + ((2b³-9abc+27a²d)/27a²)²))/2)+b/3a
((-(cbrt((-((2b³-9abc+27a²d)/27a²)+sqrt(-4(c - b²/3a)³/27 + ((2b³-9abc+27a²d)/27a²)²))/2)+cbrt((-((2b³-9abc+27a²d)/27a²)-sqrt(-4(c - b²/3a)³/27 + ((2b³-9abc+27a²d)/27a²)²))/2))±sqrt((cbrt((-((2b³-9abc+27a²d)/27a²)+sqrt(-4(c - b²/3a)³/27 + ((2b³-9abc+27a²d)/27a²)²))/2)+cbrt((-((2b³-9abc+27a²d)/27a²)-sqrt(-4(c - b²/3a)³/27 + ((2b³-9abc+27a²d)/27a²)²))/2))²-4(cbrt((-((2b³-9abc+27a²d)/27a²)+sqrt(-4(c - b²/3a)³/27 + ((2b³-9abc+27a²d)/27a²)²))/2)+cbrt((-((2b³-9abc+27a²d)/27a²)-sqrt(-4(c - b²/3a)³/27 + ((2b³-9abc+27a²d)/27a²)²))/2))))/2)+b/3a

2 months ago | [YT] | 0