{{{id=1|
## solve  
## serve per risolvere 1 equazione o una lista di equazioni (in C)
///
}}}

{{{id=2|
f = x^2+5*x+6; print f
sol = solve(x^2+5*x+6 == 0, x)
sol
///
x^2 + 5*x + 6
[x == -3, x == -2]
}}}

{{{id=7|
# verifica:
print f(x=-3)
print f.subs(x==-2)
print f.subs({x:-3})
///
0
0
0
}}}

{{{id=34|
a, b, c, d, e, f = var('a b c d e f')
///
}}}

{{{id=3|
eqz = [a*b*c+b*c*(d+e), d*(e+2*f)+a*d*(a+c), a^2-b^2, c^2-1, d^2-1, a^2*c^2-e^2]
sol = solve(eqz, a, b, c, d, e, f); sol
///
[[a == 0, b == 0, c == 1, d == 1, e == 0, f == 0], [a == 0, b == 0, c == -1, d == 1, e == 0, f == 0], [a == (-1/2), b == (1/2), c == -1, d == 1, e == (-1/2), f == (-1/8)], [a == (-1/2), b == (-1/2), c == -1, d == 1, e == (-1/2), f == (-1/8)], [a == 0, b == 0, c == 1, d == -1, e == 0, f == 0], [a == 0, b == 0, c == -1, d == -1, e == 0, f == 0], [a == (1/2), b == (-1/2), c == -1, d == -1, e == (1/2), f == (-1/8)], [a == (1/2), b == (1/2), c == -1, d == -1, e == (1/2), f == (-1/8)], [a == (-1/2), b == (1/2), c == 1, d == 1, e == (-1/2), f == (3/8)], [a == (-1/2), b == (-1/2), c == 1, d == 1, e == (-1/2), f == (3/8)], [a == (1/2), b == (-1/2), c == 1, d == -1, e == (1/2), f == (-5/8)], [a == (1/2), b == (1/2), c == 1, d == -1, e == (1/2), f == (-5/8)]]
}}}

{{{id=37|
eqz[0].subs(sol[0])
///
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_16.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("ZXF6WzBdLnN1YnMoc29sWzBdKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/tmp/tmpXzTKer/___code___.py", line 3, in <module>
    exec compile(u'eqz[_sage_const_0 ].subs(sol[_sage_const_0 ])' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "expression.pyx", line 3491, in sage.symbolic.expression.Expression.substitute (sage/symbolic/expression.cpp:15412)
TypeError: subs takes either a set of keyword arguments, a dictionary, or a symbolic relational expression
}}}

{{{id=35|
len(sol)
///
12
}}}

{{{id=10|
sol1 = solve(eqz, a, b, c, d, e, f, solution_dict=True); sol1
///
[{d: 1, c: 1, e: 0, b: 0, f: 0, a: 0}, {d: 1, c: -1, e: 0, b: 0, f: 0, a: 0}, {d: 1, c: -1, e: -1/2, b: 1/2, f: -1/8, a: -1/2}, {d: 1, c: -1, e: -1/2, b: -1/2, f: -1/8, a: -1/2}, {d: -1, c: 1, e: 0, b: 0, f: 0, a: 0}, {d: -1, c: -1, e: 0, b: 0, f: 0, a: 0}, {d: -1, c: -1, e: 1/2, b: -1/2, f: -1/8, a: 1/2}, {d: -1, c: -1, e: 1/2, b: 1/2, f: -1/8, a: 1/2}, {d: 1, c: 1, e: -1/2, b: 1/2, f: 3/8, a: -1/2}, {d: 1, c: 1, e: -1/2, b: -1/2, f: 3/8, a: -1/2}, {d: -1, c: 1, e: 1/2, b: -1/2, f: -5/8, a: 1/2}, {d: -1, c: 1, e: 1/2, b: 1/2, f: -5/8, a: 1/2}]
}}}

{{{id=36|
for i in range(len(eqz)):
    eqz[i].subs(sol1[0])
///
0
0
0
0
0
0
}}}

{{{id=38|
[eqz[i].subs(sol1[0]) for i in range(len(eqz))]
///
[0, 0, 0, 0, 0, 0]
}}}

{{{id=4|
## verifica
eqz[0].subs(sol1[0])
///
0
}}}

{{{id=39|
eqz
///
[(d + e)*b*c + a*b*c, (a + c)*a*d + (e + 2*f)*d, a^2 - b^2, c^2 - 1, d^2 - 1, a^2*c^2 - e^2]
}}}

{{{id=12|
map(lambda u: u.subs(sol1[4]), eqz)
///
[0, 0, 0, 0, 0, 0]
}}}

{{{id=13|
sol = solve(x^3+3*x+1, x); sol
///
[x == -1/2*(1/2*sqrt(5) - 1/2)^(1/3)*(I*sqrt(3) + 1) + 1/2*(-I*sqrt(3) + 1)/(1/2*sqrt(5) - 1/2)^(1/3), x == -1/2*(1/2*sqrt(5) - 1/2)^(1/3)*(-I*sqrt(3) + 1) + 1/2*(I*sqrt(3) + 1)/(1/2*sqrt(5) - 1/2)^(1/3), x == (1/2*sqrt(5) - 1/2)^(1/3) - 1/(1/2*sqrt(5) - 1/2)^(1/3)]
}}}

{{{id=14|
show(sol)
///
<html><div class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = -\frac{1}{2} \, {\left(\frac{1}{2} \, \sqrt{5} - \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} {\left(i \, \sqrt{3} + 1\right)} + \frac{-i \, \sqrt{3} + 1}{2 \, {\left(\frac{1}{2} \, \sqrt{5} - \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)}}, x = -\frac{1}{2} \, {\left(\frac{1}{2} \, \sqrt{5} - \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} {\left(-i \, \sqrt{3} + 1\right)} + \frac{i \, \sqrt{3} + 1}{2 \, {\left(\frac{1}{2} \, \sqrt{5} - \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)}}, x = {\left(\frac{1}{2} \, \sqrt{5} - \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} - \frac{1}{{\left(\frac{1}{2} \, \sqrt{5} - \frac{1}{2}\right)}^{\frac{1}{3}}}\right]</div></html>
}}}

{{{id=40|
sol
///
[x == -1/2*(1/2*sqrt(5) - 1/2)^(1/3)*(I*sqrt(3) + 1) + 1/2*(-I*sqrt(3) + 1)/(1/2*sqrt(5) - 1/2)^(1/3), x == -1/2*(1/2*sqrt(5) - 1/2)^(1/3)*(-I*sqrt(3) + 1) + 1/2*(I*sqrt(3) + 1)/(1/2*sqrt(5) - 1/2)^(1/3), x == (1/2*sqrt(5) - 1/2)^(1/3) - 1/(1/2*sqrt(5) - 1/2)^(1/3)]
}}}

{{{id=41|
sol[0].rhs()
///
-1/2*(1/2*sqrt(5) - 1/2)^(1/3)*(I*sqrt(3) + 1) + 1/2*(-I*sqrt(3) + 1)/(1/2*sqrt(5) - 1/2)^(1/3)
}}}

{{{id=42|
type(sol[0])
///
<type 'sage.symbolic.expression.Expression'>
}}}

{{{id=15|
aa = (x^3+3*x+1).subs(x=sol[0].rhs())
///
}}}

{{{id=43|
aa
///
-1/8*((1/2*sqrt(5) - 1/2)^(1/3)*(I*sqrt(3) + 1) - (-I*sqrt(3) + 1)/(1/2*sqrt(5) - 1/2)^(1/3))^3 - 3/2*(1/2*sqrt(5) - 1/2)^(1/3)*(I*sqrt(3) + 1) + 3/2*(-I*sqrt(3) + 1)/(1/2*sqrt(5) - 1/2)^(1/3) + 1
}}}

{{{id=16|
aa.simplify_full()
///
0
}}}

{{{id=17|
p, q = var('p q')
show(solve(x^3+p*x+q, x))
///
<html><div class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = \frac{{\left(-i \, \sqrt{3} + 1\right)} p}{6 \, {\left(\frac{1}{18} \, \sqrt{4 \, p^{3} + 27 \, q^{2}} \sqrt{3} - \frac{1}{2} \, q\right)}^{\left(\frac{1}{3}\right)}} - \frac{1}{2} \, {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{18} \, \sqrt{4 \, p^{3} + 27 \, q^{2}} \sqrt{3} - \frac{1}{2} \, q\right)}^{\left(\frac{1}{3}\right)}, x = \frac{{\left(i \, \sqrt{3} + 1\right)} p}{6 \, {\left(\frac{1}{18} \, \sqrt{4 \, p^{3} + 27 \, q^{2}} \sqrt{3} - \frac{1}{2} \, q\right)}^{\left(\frac{1}{3}\right)}} - \frac{1}{2} \, {\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{18} \, \sqrt{4 \, p^{3} + 27 \, q^{2}} \sqrt{3} - \frac{1}{2} \, q\right)}^{\left(\frac{1}{3}\right)}, x = -\frac{p}{3 \, {\left(\frac{1}{18} \, \sqrt{4 \, p^{3} + 27 \, q^{2}} \sqrt{3} - \frac{1}{2} \, q\right)}^{\left(\frac{1}{3}\right)}} + {\left(\frac{1}{18} \, \sqrt{4 \, p^{3} + 27 \, q^{2}} \sqrt{3} - \frac{1}{2} \, q\right)}^{\left(\frac{1}{3}\right)}\right]</div></html>
}}}

{{{id=18|
r = var('r')
f = x^4+p*x^2+q*x+r
soluz = solve(f, x)
///
}}}

{{{id=19|
len(soluz)
///
4
}}}

{{{id=20|
show(soluz[0])
///
<html><div class="math">\newcommand{\Bold}[1]{\mathbf{#1}}x = -\frac{1}{2} \, \sqrt{-\frac{4}{3} \, p - \frac{p^{2} + 12 \, r}{9 \, {\left(\frac{1}{27} \, p^{3} - \frac{4}{3} \, p r + \frac{1}{2} \, q^{2} + \frac{1}{18} \, \sqrt{4 \, p^{3} q^{2} + 128 \, p^{2} r^{2} + 27 \, q^{4} - 256 \, r^{3} - 16 \, {\left(p^{4} + 9 \, p q^{2}\right)} r} \sqrt{3}\right)}^{\left(\frac{1}{3}\right)}} + \frac{6 \, q}{\sqrt{\frac{p^{2} - 6 \, {\left(\frac{1}{27} \, p^{3} - \frac{4}{3} \, p r + \frac{1}{2} \, q^{2} + \frac{1}{18} \, \sqrt{4 \, p^{3} q^{2} + 128 \, p^{2} r^{2} + 27 \, q^{4} - 256 \, r^{3} - 16 \, {\left(p^{4} + 9 \, p q^{2}\right)} r} \sqrt{3}\right)}^{\left(\frac{1}{3}\right)} p + 12 \, r + 9 \, {\left(\frac{1}{27} \, p^{3} - \frac{4}{3} \, p r + \frac{1}{2} \, q^{2} + \frac{1}{18} \, \sqrt{4 \, p^{3} q^{2} + 128 \, p^{2} r^{2} + 27 \, q^{4} - 256 \, r^{3} - 16 \, {\left(p^{4} + 9 \, p q^{2}\right)} r} \sqrt{3}\right)}^{\left(\frac{2}{3}\right)}}{{\left(\frac{1}{27} \, p^{3} - \frac{4}{3} \, p r + \frac{1}{2} \, q^{2} + \frac{1}{18} \, \sqrt{4 \, p^{3} q^{2} + 128 \, p^{2} r^{2} + 27 \, q^{4} - 256 \, r^{3} - 16 \, {\left(p^{4} + 9 \, p q^{2}\right)} r} \sqrt{3}\right)}^{\frac{1}{3}}}}} - {\left(\frac{1}{27} \, p^{3} - \frac{4}{3} \, p r + \frac{1}{2} \, q^{2} + \frac{1}{18} \, \sqrt{4 \, p^{3} q^{2} + 128 \, p^{2} r^{2} + 27 \, q^{4} - 256 \, r^{3} - 16 \, {\left(p^{4} + 9 \, p q^{2}\right)} r} \sqrt{3}\right)}^{\left(\frac{1}{3}\right)}} - \frac{1}{6} \, \sqrt{\frac{p^{2} - 6 \, {\left(\frac{1}{27} \, p^{3} - \frac{4}{3} \, p r + \frac{1}{2} \, q^{2} + \frac{1}{18} \, \sqrt{4 \, p^{3} q^{2} + 128 \, p^{2} r^{2} + 27 \, q^{4} - 256 \, r^{3} - 16 \, {\left(p^{4} + 9 \, p q^{2}\right)} r} \sqrt{3}\right)}^{\left(\frac{1}{3}\right)} p + 12 \, r + 9 \, {\left(\frac{1}{27} \, p^{3} - \frac{4}{3} \, p r + \frac{1}{2} \, q^{2} + \frac{1}{18} \, \sqrt{4 \, p^{3} q^{2} + 128 \, p^{2} r^{2} + 27 \, q^{4} - 256 \, r^{3} - 16 \, {\left(p^{4} + 9 \, p q^{2}\right)} r} \sqrt{3}\right)}^{\left(\frac{2}{3}\right)}}{{\left(\frac{1}{27} \, p^{3} - \frac{4}{3} \, p r + \frac{1}{2} \, q^{2} + \frac{1}{18} \, \sqrt{4 \, p^{3} q^{2} + 128 \, p^{2} r^{2} + 27 \, q^{4} - 256 \, r^{3} - 16 \, {\left(p^{4} + 9 \, p q^{2}\right)} r} \sqrt{3}\right)}^{\frac{1}{3}}}}</div></html>
}}}

{{{id=21|
## (f.subs(x=soluz[0].rhs())).simplify_full()
///
}}}

{{{id=22|
x, y, z = var('x, y, z')
solve([x^2+y^2+z^2-4, x^2+2*y+3*z==4, y-3*x+z^2+1], x, y, z)
///
[[x == (3.37946086473 - 1.9907019517*I), y == (2.80998440395 + 4.20502937165*I), z == (-3.02594342789 + 1.68164664497*I)], [x == (3.37946086473 + 1.9907019517*I), y == (2.80998440395 - 4.20502937165*I), z == (-3.02594342789 - 1.68164664497*I)], [x == (-2.79854050905 - 1.27846700704*I), y == (-2.29312418563 + 0.5943671272*I), z == (0.796299092861 - 2.78147255726*I)], [x == (-2.79854050905 + 1.27846700704*I), y == (-2.29312418563 - 0.5943671272*I), z == (0.796299092861 + 2.78147255726*I)], [x == 0.928026172301, y == 1.76676328502, z == -0.131586362356], [x == 0.910133136095, y == -0.800483675937, z == 1.59087509349], [x == 1/2*I*sqrt(51) - 3/2, y == 5, z == 1/2*I*sqrt(3)*sqrt(17) + 3/2], [x == -1/2*I*sqrt(51) - 3/2, y == 5, z == -1/2*I*sqrt(3)*sqrt(17) + 3/2]]
}}}

{{{id=28|
p1 = implicit_plot3d(x^2+y^2+z^2-4, (x, -3, 3), (y, -3, 3), (z, -3, 3), rgbcolor=(1, 0, 0))
p2 = implicit_plot3d(x^2+2*y+3*z==4, (x, -3, 3), (y, -3, 3), (z, -3, 3), rgbcolor=(1, 1, 0))
p3 = implicit_plot3d(y-3*x+z^2+1, (x, -3, 3), (y, -3, 3), (z, -3, 3), rgbcolor=(0, 1, 1))
///
}}}

{{{id=44|
#show(p1+p2+p3)
///
}}}

{{{id=23|
x, y, z, t = var("x y z t")
solve([x^2+y^2+z^2-1, y^2+z^2+t^2-1, x*y*z*t], x, y, z, t)
///
[[x == 0, y == r1, z == sqrt(-r1^2 + 1), t == 0], [x == 0, y == r2, z == -sqrt(-r2^2 + 1), t == 0], [x == r3, y == sqrt(-r3^2 + 1), z == 0, t == -r3], [x == r4, y == sqrt(-r4^2 + 1), z == 0, t == r4], [x == r5, y == -sqrt(-r5^2 + 1), z == 0, t == -r5], [x == r6, y == -sqrt(-r6^2 + 1), z == 0, t == r6], [x == r7, y == 0, z == sqrt(-r7^2 + 1), t == -r7], [x == r8, y == 0, z == sqrt(-r8^2 + 1), t == r8], [x == r9, y == 0, z == -sqrt(-r9^2 + 1), t == -r9], [x == r10, y == 0, z == -sqrt(-r10^2 + 1), t == r10]]
}}}

{{{id=24|
### approssimazioni numeriche di soluzioni
///
}}}

{{{id=25|
f = sin(3*x)-cos(x)
plot(f, (x, -3, 3), figsize=[4, 4])
///
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=26|
find_root(f, -2, -1)
///
-1.1780972450962113
}}}

{{{id=27|
find_root(f, 1.5, 2)
///
1.9634954084936209
}}}

{{{id=32|

///
}}}