HP 33s hp 33s_user's manual_English_E_HDPM20PIE56.pdf - Page 243
is solved by the formula, A quadratic equation
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b0 = a0(4a2 - a32) - a12. Let y0 be the largest real root of the above cubic. Then the fourth-order polynomial is reduced to two quadratic polynomials: x2 + (J + L)x + (K + M) = 0 x2 + (J - L)x + (K - M) = 0 where J = a3/2 K = y0 /2 L = J2 − a2 + y0 × (the sign of JK - a1/2) M = K 2 − a0 Roots of the fourth degree polynomial are found by solving these two quadratic polynomials. A quadratic equation x2 + a1x + a0 = 0 is solved by the formula x1,2 = − a1 2 ± ( a1 )2 2 − a0 If the discriminant d = (a1/2)2 - ao ≥ 0, the roots are real; if d < 0, the roots are complex, being u ± iv = −(a1 2) ± i − d . Mathematics Programs 15-21
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Mathematics Programs
15–21
b
0
=
a
0
(4
a
2
–
a
3
2
) –
a
1
2
.
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
.