The Cubic Formula and Derivation
The Cubic Formula and Derivation
Daniel Rui
Here is the general cubic, with the x3 coefficient already divided into the other coefficients, right
hand side already set to zero because we are finding roots: x3 + ax2 + bx + c = 0.
We
substitute
in
x
=
y
-
a 3
to
get
( y3
-
2a y2
+
a2
y
-
a y2
+
2a2
y
-
a3
)
3
93
9 27
+a
( y2
-
2a y
+
a2
)
39
( a) +b y -
3
+c = 0
Simplifying to give
y3-ay2+ a2 y- a3 3 27
+ay2- 2a2 y+ a3 39 ab +by- 3
+c = 0
Simplifying more to give
y3
-
a2 y
+
by
-
a3
+
a3
-
ab
+c
=
0
3
27 9 3
This is of the form y3 + dy + e = 0 where
a2
a3 a3 ab
d = b - and e = - + - + c
3
27 9 3
1
We substitute y = 3 u - 3 v to get
3 u3
-
3
3 u2
3v
+
3
3u
3 v2
-
3v
3
+
d3u
-
d3v
+
e
=
0
Simplifying to get
(u
-
v)
-
d3v
-
33 u2
3v
+
d3u
+
3
3u
3 v2
+
e
=
0
Anti-distributing to get
(u
-
v)
-
3 v(d
-
33 u2)
+
3 u(d
+
33 v2)
+
e
=
0
Let's
define
v
-
u
=
e
('cause
we
can),
causing
- 3 v(d
-
33 u2)
+
3 u(d
+
33 v2)
=
0,
which
can
be
simplified to
3 u(d
+
33 v2)
=
3 v(d
+
33 u2)
d3u
+
3
3u
3 v2
=
d3v
+
3
3v
3 u2
d3u
-
d3v
=
3
3v
3 u2
-
3
3u
3v
2
d( 3 u - 3 v) = 3 3 v 3 u( 3 u - 3 v)
d=33v3u
d = 3 uv
3 d3
= uv 27
From the first purple equation, we have v = e + u, which we can put into the second to get
u(e + u) = d3 u2 + eu - d3 = 0
27
27
2
Where we can solve for u with the quadratic formula.
-e ? u=
e2
+
4
d3 27
2
We can also get equations for v; the first equation gives u = v - e, which we stuff into the second
equation to get
v(v - e) = d3 v2 - ev - d3
27
27
Quadratic formula yielding
e? v=
e2
+
4
d3 27
2
Via our definition above (y = 3 u - 3 v), we get
3 -e ? y=
e2
+
4
d3 27
-
3
e?
2
e2
+
4
d3 27
2
And
from
the
definition
above
that
(x
=
y
-
a 3
),
we
have
3 -e ? x=
e2
+
4
d3 27
-
3
e?
2
e2
+
4
d3 27
-
a
2
3
However, there is a problem. The ? gives solutions that don't satisfy e = v - u, so we just keep the positive. And finally, if we want, we can plug in d and e to get the cubic formula
x=
3
-(-
a3 27
+
a3 9
-
ab 3
+ c) +
(-
a3 27
+
a3 9
-
ab 3
+ c)2
+
4
(b-
a2 3
27
)3
2
3
-
(-
a3 27
+
a3 9
-
ab 3
+
c)
+
(-
a3 27
+
a3 9
-
ab 3
+
c)2
+
4 (b-
a2 3
)3
27
a -
2
3
This formula only gives one root; using roots of unity we can get the others.
3
................
................
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