modifica
a
x
+
b
=
0
;
x
=
−
b
a
;
{\displaystyle ax+b=0\ ;\qquad x=-{b \over a}\ ;}
determinata se
a
≠
0
;
{\displaystyle \ a\neq 0;}
a
=
b
=
0
;
{\displaystyle \ a=b=0;}
a
=
0
,
b
≠
0.
{\displaystyle \ a=0,\ b\neq 0.}
modifica
a
x
2
+
b
x
+
c
=
0
;
{\displaystyle ax^{2}+bx+c=0;}
x
=
−
b
±
b
2
−
4
a
c
2
a
,
o
v
v
e
r
o
:
x
=
−
k
±
k
2
−
a
c
a
(
k
=
b
2
)
;
{\displaystyle \ x={-b\pm {\sqrt {b^{2}-4ac}} \over 2a},\qquad ovvero:\qquad x={-k\pm {\sqrt {k^{2}-ac}} \over a}\qquad (k={b \over 2});}
le due radici sono reali distinte se
b
2
−
4
a
c
>
0
,
{\displaystyle \ b^{2}-4ac>0,}
le due radici sono reali coincidenti se
b
2
−
4
a
c
=
0
,
{\displaystyle \ b^{2}-4ac=0,}
le due radici sono complesse coniugate se
b
2
−
4
a
c
<
0.
{\displaystyle \ b^{2}-4ac<0.}
modifica
a
x
3
+
b
x
2
+
c
x
+
d
=
0
{\displaystyle ax^{3}+bx^{2}+cx+d=0}
a) Ponendo
x
=
y
−
b
3
a
{\displaystyle \ x=y-{b \over 3a}}
y
3
+
p
y
+
y
=
0
{\displaystyle \ y^{3}+py+y=0}
da cui
:
y
=
−
q
2
+
q
2
4
+
p
3
27
3
+
−
q
2
−
q
2
4
+
p
3
27
3
{\displaystyle :\qquad y={\sqrt[{3}]{-{q \over 2}+{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}+{\sqrt[{3}]{-{q \over 2}-{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}}
(formula risolutiva Cardanica
se
q
2
4
+
p
3
27
>
0
{\displaystyle {q^{2} \over 4}+{p^{3} \over 27}>0}
reale complesse coniugate se
q
2
4
+
p
3
27
=
0
{\displaystyle {q^{2} \over 4}+{p^{3} \over 27}=0}
reale semplice doppia se
q
2
4
+
p
3
27
<
0
{\displaystyle {q^{2} \over 4}+{p^{3} \over 27}<0}
reali distinte In quest'ultimo caso le tre radici si possono calcolare agevolmente in forma trigonometrica. Ponendo:
−
q
2
+
i
−
(
q
2
4
+
p
3
27
)
2
=
ρ
(
cos
θ
+
i
sin
θ
)
{\displaystyle {-q \over 2}+i{\sqrt[{2}]{-({q^{2} \over 4}+{p^{3} \over 27})}}=\rho (\cos \theta +i\sin \theta )}
−
q
2
−
i
−
(
q
2
4
+
p
3
27
)
2
=
ρ
(
cos
θ
−
i
sin
θ
)
{\displaystyle {-q \over 2}-i{\sqrt[{2}]{-({q^{2} \over 4}+{p^{3} \over 27})}}=\rho (\cos \theta -i\sin \theta )}
dove:
ρ
=
−
p
3
27
2
cos
θ
=
−
q
2
ρ
θ
=
a
r
c
cos
(
−
q
2
ρ
)
{\displaystyle \rho ={\sqrt[{2}]{-p^{3} \over 27}}\qquad \cos \theta ={-q \over 2\rho }\qquad \theta =arc\cos({-q \over 2\rho })}
si ha allora:
α
1
=
ρ
3
(
cos
θ
3
+
i
sin
θ
3
)
,
β
1
=
ρ
3
(
cos
θ
3
−
i
sin
θ
3
)
;
{\displaystyle \alpha _{1}={\sqrt[{3}]{\rho }}(\cos {\theta \over 3}+i\sin {\theta \over 3}),\qquad \beta _{1}={\sqrt[{3}]{\rho }}(\cos {\theta \over 3}-i\sin {\theta \over 3});}
α
2
=
ρ
3
(
cos
θ
+
2
π
3
+
i
sin
θ
+
2
π
3
)
,
z
)
{\displaystyle \alpha _{2}={\sqrt[{3}]{\rho }}(\cos {\theta +2\pi \over 3}+i\sin {\theta +2\pi \over 3}),\qquad z)}
b) Si può determinare per tentativi o con approssimazioni successive il valore di una radice servendosi dell'osservazione:
se
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
{\displaystyle \ f(x)=ax^{3}+bx^{2}+cx+d}
e
f
(
a
)
=
k
2
,
{\displaystyle \ f(a)=k^{2},}
f
(
b
)
=
−
h
2
,
{\displaystyle \ f(b)=-h^{2},}
a
{\displaystyle \ a}
b
{\displaystyle \ b}
m
{\displaystyle \ m}
0
{\displaystyle \ 0}
f
(
x
)
x
−
m
{\displaystyle {\frac {f(x)}{x-m}}}
x
{\displaystyle \ x}
In particolare le equazioni reciproche:
a
x
3
+
b
x
2
±
b
x
±
a
=
0
{\displaystyle \ ax^{3}+bx^{2}\pm bx\pm a=0}
ammettono la radice
x
=
∓
1
;
{\displaystyle \ x=\mp 1;}
a
x
2
+
(
b
∓
a
)
x
+
a
=
0.
{\displaystyle \ ax^{2}+(b\mp a)x+a=0.}
c) L'equazione
x
3
+
p
x
+
q
=
0
{\displaystyle \ x^{3}+px+q=0}
graficamente
y
=
x
3
{\displaystyle \ y=x^{3}}
le radici cercate sranno le ascisse dei punti comuni alla parabola cubica:
y
=
x
3
{\displaystyle \ y=x^{3}}
retta:
y
+
p
x
+
q
=
0.
{\displaystyle \ y+px+q=0.}
modifica
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
c
=
0.
{\displaystyle \ ax^{4}+bx^{3}+cx^{2}+dx+c=0.}
Ponendo
x
=
y
−
b
4
a
{\displaystyle \ x=y-{b \over 4a}}
y
4
+
p
x
2
+
q
y
+
r
=
0.
{\displaystyle \ y^{4}+px^{2}+qy+r=0.}
Se
q
=
0
,
{\displaystyle \ q=0,}
biquadratica
y
2
=
z
.
{\displaystyle \ y^{2}=z.}
Se
q
≠
0
,
{\displaystyle \ q\neq 0,}
y
=
u
+
v
+
w
,
{\displaystyle \ y=u+v+w,}
u
2
,
v
2
,
w
2
{\displaystyle \ u^{2},v^{2},w^{2}}
z
3
+
p
2
z
2
+
(
p
2
16
−
r
4
)
z
−
q
2
64
=
0.
{\displaystyle \ z^{3}+{p \over 2}z^{2}+({p^{2} \over 16}-{r \over 4})z-{q^{2} \over 64}=0.}
Se
γ
,
μ
,
∋
{\displaystyle \ \gamma ,\mu ,\ni }
u
=
±
γ
2
;
v
=
±
μ
2
;
w
=
±
ν
2
.
{\displaystyle \ u=\pm {\sqrt[{2}]{\gamma }};\qquad v=\pm {\sqrt[{2}]{\mu }};\qquad w=\pm {\sqrt[{2}]{\nu }}.}
Si sceglieranno poi tre fra i valori di
u
,
v
,
w
{\displaystyle \ u,v,w}
u
v
w
=
−
q
8
,
{\displaystyle uvw=-{q \over 8},}
e se
γ
2
,
μ
2
,
n
u
2
{\displaystyle \ {\sqrt[{2}]{\gamma }},{\sqrt[{2}]{\mu }},{\sqrt[{2}]{nu}}}
y
1
=
γ
2
+
μ
2
+
ν
2
,
y
2
=
γ
2
−
μ
2
−
ν
2
,
{\displaystyle \ y_{1}={\sqrt[{2}]{\gamma }}+{\sqrt[{2}]{\mu }}+{\sqrt[{2}]{\nu }},\qquad y_{2}={\sqrt[{2}]{\gamma }}-{\sqrt[{2}]{\mu }}-{\sqrt[{2}]{\nu }},}
y
3
=
−
γ
2
+
μ
2
−
ν
2
,
y
4
=
−
γ
2
−
μ
2
+
ν
2
.
{\displaystyle \ y_{3}=-{\sqrt[{2}]{\gamma }}+{\sqrt[{2}]{\mu }}-{\sqrt[{2}]{\nu }},\qquad y_{4}=-{\sqrt[{2}]{\gamma }}-{\sqrt[{2}]{\mu }}+{\sqrt[{2}]{\nu }}.}
L'equazione può avere 4 radici reali; reali immaginarie; immaginarie;
z
.
{\displaystyle \ z.}
a) Esponenziale monomia:
b
f
(
x
)
=
a
,
{\displaystyle \ b^{f(x)}=a,}
f
(
x
)
{\displaystyle \ f(x)}
f
(
x
)
=
l
o
g
b
a
=
L
o
g
a
L
o
g
b
=
l
o
g
a
l
o
g
b
{\displaystyle \ f(x)=log_{b}\ a={Log\ a \over Log\ b}={log\ a \over log\ b}}
b) Esponenziale trinomia:
a
k
2
x
+
b
k
x
+
c
=
0.
{\displaystyle \ ak^{2x}+bk^{x}+c=0.}
Ponendo:
y
=
k
x
,
{\displaystyle \ y=k^{x},}
x
=
log
k
y
,
{\displaystyle \ x=\log _{k}y,}
y
.
{\displaystyle \ y.}
c) Logaritmica:
log
b
f
(
x
)
±
log
b
ϕ
(
x
)
=
a
{\displaystyle \log _{b}f(x)\pm \log _{b}\phi (x)=a}
Se
f
(
x
)
{\displaystyle \ f(x)}
ϕ
(
x
)
{\displaystyle \ \phi (x)}
f
(
x
)
ϕ
(
x
)
=
b
a
f
(
x
)
ϕ
(
x
)
=
b
a
.
{\displaystyle \ f(x)\phi (x)=b^{a}\qquad {f(x) \over \phi (x)}=b^{a}.}
d) Trigonometrica lineare in
sin
x
,
cos
x
:
{\displaystyle \ \sin x,\cos x:}
a
sin
x
+
b
cos
x
=
c
.
{\displaystyle \ a\sin x+b\cos x=c.}
con la trasformazione:
sin
x
=
2
tan
x
2
1
+
tan
2
x
2
,
cos
x
=
1
−
tan
2
x
2
1
+
tan
2
x
2
{\displaystyle \sin x={2\tan {x \over 2} \over 1+\tan ^{2}{x \over 2}},\qquad \cos x={1-\tan ^{2}{x \over 2} \over 1+\tan ^{2}{x \over 2}}}
l'equazione diventa algebrica in
tan
x
2
.
{\displaystyle \tan {x \over 2}.}
e) Trigonometrica omogenea di 2° in
sin
x
,
cos
x
:
{\displaystyle \ \sin x,\cos x:}
a
sin
2
x
+
b
sin
x
cos
x
+
c
cos
2
x
=
0
{\displaystyle \ a\sin ^{2}x+b\sin x\cos x+c\cos ^{2}x=0}
Dividendo per
cos
2
x
,
{\displaystyle \ \cos ^{2}x,}
tan
x
;
{\displaystyle \ \tan x;}
a
sin
2
x
+
b
sin
x
cos
x
+
c
cos
2
x
=
d
,
{\displaystyle \ a\sin ^{2}x+b\sin x\cos x+c\cos ^{2}x=d,}
ponendo:
d
=
d
(
sin
2
x
+
cos
2
x
)
.
{\displaystyle \ d=d(\sin ^{2}x+\cos ^{2}x).}
f) Trigonometrica razionale intera
tan
x
,
cot
x
:
{\displaystyle \ \tan x,\cot x:}
f
(
tan
x
,
cot
x
)
=
0.
{\displaystyle \ f(\tan x,\cot x)=0.}
Si riduce algebrica razionale intera in
tan
x
{\displaystyle \ \tan x}
cot
x
=
1
tan
x
.
{\displaystyle \ \cot x={1 \over \tan x}.}
![{\displaystyle :\qquad y={\sqrt[{3}]{-{q \over 2}+{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}+{\sqrt[{3}]{-{q \over 2}-{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2bc3b9c8178f0f73ab3dae951351176d07fff1)
![{\displaystyle {-q \over 2}+i{\sqrt[{2}]{-({q^{2} \over 4}+{p^{3} \over 27})}}=\rho (\cos \theta +i\sin \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b242289458e954f7ed3b87aaf74522f774df3bf)
![{\displaystyle {-q \over 2}-i{\sqrt[{2}]{-({q^{2} \over 4}+{p^{3} \over 27})}}=\rho (\cos \theta -i\sin \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/523490a54b087be06dd6bc088eba4731d8be1293)
![{\displaystyle \rho ={\sqrt[{2}]{-p^{3} \over 27}}\qquad \cos \theta ={-q \over 2\rho }\qquad \theta =arc\cos({-q \over 2\rho })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70cca823161bc69d47775496f8f14dd5adb383cd)
![{\displaystyle \alpha _{1}={\sqrt[{3}]{\rho }}(\cos {\theta \over 3}+i\sin {\theta \over 3}),\qquad \beta _{1}={\sqrt[{3}]{\rho }}(\cos {\theta \over 3}-i\sin {\theta \over 3});}](https://wikimedia.org/api/rest_v1/media/math/render/svg/490749df2a682daf674ddb4697ed87b0f0884252)
![{\displaystyle \alpha _{2}={\sqrt[{3}]{\rho }}(\cos {\theta +2\pi \over 3}+i\sin {\theta +2\pi \over 3}),\qquad z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9ca39c315b71d9f9ee238b12e65652086b53ae)
![{\displaystyle \ u=\pm {\sqrt[{2}]{\gamma }};\qquad v=\pm {\sqrt[{2}]{\mu }};\qquad w=\pm {\sqrt[{2}]{\nu }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ade18f87d86686b3746268cadb1260427d33977c)
![{\displaystyle \ {\sqrt[{2}]{\gamma }},{\sqrt[{2}]{\mu }},{\sqrt[{2}]{nu}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d22eb32d1e1032f317a737c22df613444c15db3)
![{\displaystyle \ y_{1}={\sqrt[{2}]{\gamma }}+{\sqrt[{2}]{\mu }}+{\sqrt[{2}]{\nu }},\qquad y_{2}={\sqrt[{2}]{\gamma }}-{\sqrt[{2}]{\mu }}-{\sqrt[{2}]{\nu }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1794b4029565a3877016d830bcd773f707101af3)
![{\displaystyle \ y_{3}=-{\sqrt[{2}]{\gamma }}+{\sqrt[{2}]{\mu }}-{\sqrt[{2}]{\nu }},\qquad y_{4}=-{\sqrt[{2}]{\gamma }}-{\sqrt[{2}]{\mu }}+{\sqrt[{2}]{\nu }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3568a1dd29ea417f51b3083c5e44a35cc1aa0ecc)
![{\displaystyle \ f(x,{\sqrt[{n}]{\phi (x)}})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d243956d929ec1fbb3874d2ec50cbffd647fab2)
