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PROGRAM
T I
TL
E
QUADRATIC
AND
CUBIC E
QUATIONS
(3) Imaginary root with b
2
- 4ac < O
- b
Real
part
:~
J4ac b
2
Imaginary part : -
2a
II
) Root
ofa
cubic equation:
Cardano's method is
used for a solution.
PROGRAM
NO.
PS
-
A-6
AX
3
+
BX
2
+ CX
+D
= 0
(A
'l\
0 and all factors are real numbers.)
The following is obtained
by dividing factors by A:
x
3
+ ax
2
+bx+
c = O
Through conversion
of
y = x -
~
3 •
y' - 3 PY
-q
~
0
b a'
p
c--
-
3 9
let
Y = u +
v,
then the following is obtained:
MS T
V)
.-
3 U \ ' ( U 't
t•
) - 3 P ( U - \ ' ) T 4 = 0
Let u
3
+p
3
= -
q,
then
uv
= -
p.
ti
, +
\13
=-
q,
u' v
,=
-
Pl
This shows
that
u
3
and v
3
are the roots
of
a quadratic equation
of
t
2
+
qt
- p
3
= 0.
That
is to say,
..
•=
~
(- ¢+/ q' +
4p
')
•' ::
.!..
(- q- f ¢
1
+
4p
')
2
From this, the r
oo
ts
a,
Jl
and
'Y
of
y
3
+ 3 py ·+ q = 0
become as follows:
Th
e above are
to
be divided into the real and imaginary parts.
2
(I) When q
2
+
4p
3
>
0,
u
3
and v
3
are real numbers.
Therefore, u and v are the real cubic
root
of
u
3
and v
3
,
and
th
e above
formula can be used as
it
is.
This is the case
for
one real root and two
imaginary roots.
Do
not
sale this PDF !!!
-
19
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