HP 33s - Manual (Page 243)

Mathematics Programs 15–21
b
0
= a
0
(4a
2
– a
32
) – a
12
.
Let y
0
be the largest real root of the above cubic. Then the fourth–order polynomial
is reduced to two quadratic polynomials:
x
2
+ (J + L)x + (K + M) = 0
x
2
+ (J – L)x + (K – M) = 0
where J = a
3
/2
K = y
0
/2
L =
0 2
2 y a J + − × (the sign of JK – a
1
/2)
M =
0
2 a K −
Roots of the fourth degree polynomial are found by solving these two quadratic
polynomials.
A quadratic equation x
2
+ a
1
x + a
0
= 0 is solved by the formula
0
2 1 1
2 , 1
)
2
(
2
a
a a
x − ± − =
If the discriminant d = (a
1
/2)
2
– a
o
≥ 0, the roots are real; if d < 0, the roots are
complex, being d i a iv u − ± − = ± ) 2 (
1
.