86
ON CUBIC EQUATIONS
TRACT 28
Forms ofcases.
M
0 •
j3 «>
Values ofe.
Forms of the equation.
+ M
|rv'+ra
, 71 + 3
•T 3 —- r~x
4
"—1 3
4
1
+ 12
^ 4 /+ 12
jr 3 — tj- 5 Ax
=
- yj- 3
2
+ 11
+V + U
x l — l 4 r’x
=
1 o v 3T 7
3
+ 10
P'\/ + 10
x 3 — + r\r
=
- $r 3
4
+ 9
+V+9
x 3 — i^Ax
=
- $A
5
+8
\S ~b^
x 1 — + ?+r
—
__ 7 -)-*34 1
6
+7
t^’v 7 +7
A’ 3 — >+ +.r
=
-
7
+ 6
tv+6
.r 3 — %Ax
=
5 <y» 3
8
+.5
t V +5
x 3 — fAx
=
_ 4 7*3
T'
9
+4
tv+4
x 3 — Ax
=
- |r 3
10
+ 3
fV + 3
x 3 — ~Ax
—
_| r3
11
+2
fjV + 2
x 3 - \Ax
=
-V
12
+ 1
tV + i
x 3 — -*Ax
=
13
±0
fV ±0
X 3 —
=
+ i 4 ’ 3
14
— 1
tv — 1
x 3 — fAx
=
+ f 3J
15
-2
tv —2
x 3 —
=
+ ir 3
16
-3
tv -3
x 3 qp f Ax
—
+ p' 3
17
-4
tv -4
x 3 + tV
=
+ P 3
IS
— 5
t V - 5
.r 3 + |»' 2 a’
=
+ P 3
19
-6
tv - 6
x 3 + ^Ax
=
+ {A
20
-7
tv -7
x 3 + -\Ax
+ Sp3
21
-8
£?V -8
.r 3 + {Ax
-
+ -Jr 3
22
-9
tv— 9
x 3 + %Ax
='
+ v°r 3
23
-10
tV -10
x 3 + %Ax
—
+ V? .3
24
-11
tV - 11
x 3 + %Ax
=
+ \f Z A
25
— 12
tv -12
x 3 + -Jr".r
=
+ V 3 /’ 3
— n
tv~ n
7 +—r-r
. 71 +1 ,= + __ r
Values of the root r, viz.r =
-Ti — +
n+3.q _
n^l.p — V\5qlip14 q
10 b1 3</
"s>
1 ‘2 if8p
II?
10 7~6p9qbp
4p
Jq3p_Gj2f
Jl
ip
4qOp3q_
"PI?
3p
_o»
4p
_ 2 ?
5p
-i
"fiB3q_
7p
4?
'8 P5qTp_^7lOp
t?
lip
8 ?
12p
<>±v.,pn — 3.q
n + 3
=
v 1514
Ap
1 4/77:
1 j
_ 3, 4 7
n/?^11
v 103/il
= :V 4 i
12
:+ 4P =11
4/
, 4 9
-
10
=
9
= 4/
4p
= + =
V =
V =
- / 4p —
- 4/ T -
- v/^!i —" v 3 —
= 4/f
4/)
z *y — ~
v 0
:v 'f =
‘*4■’'? =v=
=*? =
iv-te =
6
= ✓* =
:</' =
V 8
At>
^ 9 - =
3 4 V
^7
^7
V 4 !4 ?1
3/?
V 0
3/?
4 / T
3 /l_7 3i, 4 9
^7
>/?
3/7
^6
3/?.
4/7
3 4 ?
47 8
* y l 2
4/y
3yl?^10, 4 7iT
4/
_ 4£ _ 4
V —
13
« + 1
= ✓
_4£_
n—2
_ 3 .J 7'''» + 1
Values of thetwo otherroots, viz.
— fr x
1 ± \/ +
1 ±++I 2
1 + + + 11
l±v/-f 10
l±4/+9
l±4/+8
l±4/+7
l±4/ + 6
1 ± + +5l±4/ + 41±4/.f 3
l± + + 2
1±4/ + 1
l±4/±01+4/-11 ± 4 / — 2
1 + 4/-31 ± + — 4
1 ± v — r>
l±v /-61 ± 4 / - 71 ± 4/ — 8i±4/-9I ± + -101 + + -111 + 4/-121 ± 4 / —71