Sharp PC-1500 Applications Manual page 25

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PROGRAM
T I TL E
QUADRATIC AND CUBIC EQUATIONS
(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:
M S 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 roots a,
Jl
and
'Y
of y
3
+ 3
py
·+ q
=
0
become as follows:
The 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 the above
formula can be used as it is. This is the case for one real root and two
imaginary roots.
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