Sharp PC-1500 - Page 26

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P ROG RAM
TIT LE
'
QU ADRA TIC A ND CUBIC EQUAT IO NS
(2) Wh en q
2
+ 4p
3
< 0, u
3
a nd v
3
ar c imaginary r oo ts.
Let u
3 = rci e then v
3
= re- 10,
r = _ p3
J- ql - 4 pJ
8 = tan
1
therefore,
-q
,.--,.,...( 9 . . 9 )
1i =J 1· c-o s3 - ls1 n3
"- r=JT" (c ·o• ~ · i • in ~ )
Through this, roots a , /3 and 'Y of Y
3
+ 3PY + q = 0
arc as follows:« = -2.[T, in ( ..'.:.-~ )
2 3
::' f)
/J =- 2 J-f' s i" (6 1" 3 )
~ c .::. __ o l
·y =- 2 J - r s i n
6 3
This is the case of thr ee different real ro ots.
P ROG RAM NO.
PS- A- 6
(3) When p
1
+4 p
3
= O and p '\ 0, u
3
= v
3
= - i is obtained.
Th er efo re, from U = V = V-i
°' = 2u
f3='Y= - u
This is the case of a double root and a another root.
(4) If q
2
+ 4p
3
= O and p = 0, q = 0 is o bt ained. The refore,
since u
= v = 0, the result is;
a=f3 = 'Y = 0
This is a triple root.
Adding
~ to °'• {3 and 'Y fi nds solutions to the e qu ation.
Be noted
that wh en 2 or 3 different roots arc very close to each other, they
may
be re ga r ded as a d ouble or t ripl e root and vice versa.
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