def test_roots_quadratic():
assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [-Rational(3, 2), 0]
assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2]
assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2]
f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c)
assert roots_quadratic(Poly(f, x)) == \
[-e*(a + c)/(a - c) - sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2),
-e*(a + c)/(a - c) + sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2)]
# check for simplification
f = Poly(y*x**2 - 2*x - 2*y, x)
assert roots_quadratic(f) == \
[-sqrt(2*y**2 + 1)/y + 1/y, sqrt(2*y**2 + 1)/y + 1/y]
f = Poly(x**2 + (-y**2 - 2)*x + y**2 + 1, x)
assert roots_quadratic(f) == \
[y**2/2 - sqrt(y**4)/2 + 1, y**2/2 + sqrt(y**4)/2 + 1]
f = Poly(sqrt(2)*x**2 - 1, x)
r = roots_quadratic(f)
assert r == _nsort(r)
# issue 8255
f = Poly(-24*x**2 - 180*x + 264)
assert [w.n(2) for w in f.all_roots(radicals=True)] == \
[w.n(2) for w in f.all_roots(radicals=False)]
for _a, _b, _c in cartes((-2, 2), (-2, 2), (0, -1)):
f = Poly(_a*x**2 + _b*x + _c)
roots = roots_quadratic(f)
assert roots == _nsort(roots)
def test_roots_quadratic():
assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [Rational(-3, 2), 0]
assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2]
assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2]
_check(Poly(2*x**2 + 4*x + 3, x).all_roots())
f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c)
assert roots_quadratic(Poly(f, x)) == \
[-e*(a + c)/(a - c) - sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2))/(a - c),
-e*(a + c)/(a - c) + sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2))/(a - c)]
# check for simplification
f = Poly(y*x**2 - 2*x - 2*y, x)
assert roots_quadratic(f) == \
[-sqrt(2*y**2 + 1)/y + 1/y, sqrt(2*y**2 + 1)/y + 1/y]
f = Poly(x**2 + (-y**2 - 2)*x + y**2 + 1, x)
assert roots_quadratic(f) == \
[1,y**2 + 1]
f = Poly(sqrt(2)*x**2 - 1, x)
r = roots_quadratic(f)
assert r == _nsort(r)
# issue 8255
f = Poly(-24*x**2 - 180*x + 264)
assert [w.n(2) for w in f.all_roots(radicals=True)] == \
[w.n(2) for w in f.all_roots(radicals=False)]
for _a, _b, _c in cartes((-2, 2), (-2, 2), (0, -1)):
f = Poly(_a*x**2 + _b*x + _c)
roots = roots_quadratic(f)
assert roots == _nsort(roots)
def _solve_as_poly(f, symbol, solveset_solver, invert_func):
"""
Solve the equation using polynomial techniques if it already is a
polynomial equation or, with a change of variables, can be made so.
"""
result = None
if f.is_polynomial(symbol):
solns = roots(f, symbol, cubics=True, quartics=True,
quintics=True, domain='EX')
num_roots = sum(solns.values())
if degree(f, symbol) <= num_roots:
result = FiniteSet(*solns.keys())
else:
poly = Poly(f, symbol)
solns = poly.all_roots()
if poly.degree() <= len(solns):
result = FiniteSet(*solns)
else:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
else:
poly = Poly(f)
if poly is None:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
gens = [g for g in poly.gens if g.has(symbol)]
if len(gens) == 1:
poly = Poly(poly, gens[0])
gen = poly.gen
deg = poly.degree()
poly = Poly(poly.as_expr(), poly.gen, composite=True)
poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True,
quintics=True).keys())
if len(poly_solns) < deg:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
if gen != symbol:
y = Dummy('y')
lhs, rhs_s = invert_func(gen, y, symbol)
if lhs is symbol:
result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
else:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
else:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
if result is not None:
if isinstance(result, FiniteSet):
# this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
# - sqrt(2)*I/2. We are not expanding for solution with free
# variables because that makes the solution more complicated. For
# example expand_complex(a) returns re(a) + I*im(a)
if all([s.free_symbols == set() and not isinstance(s, RootOf)
for s in result]):
s = Dummy('s')
result = imageset(Lambda(s, expand_complex(s)), result)
return result
else:
return ConditionSet(symbol, Eq(f, 0), S.Complexes)
def _solve_as_poly(f, symbol, solveset_solver, invert_func):
"""
Solve the equation using polynomial techniques if it already is a
polynomial equation or, with a change of variables, can be made so.
"""
result = None
if f.is_polynomial(symbol):
solns = roots(f, symbol, cubics=True, quartics=True,
quintics=True, domain='EX')
num_roots = sum(solns.values())
if degree(f, symbol) <= num_roots:
result = FiniteSet(*solns.keys())
else:
poly = Poly(f, symbol)
solns = poly.all_roots()
if poly.degree() <= len(solns):
result = FiniteSet(*solns)
else:
result = ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes)
else:
poly = Poly(f)
if poly is None:
result = ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes)
gens = [g for g in poly.gens if g.has(symbol)]
if len(gens) == 1:
poly = Poly(poly, gens[0])
gen = poly.gen
deg = poly.degree()
poly = Poly(poly.as_expr(), poly.gen, composite=True)
poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True,
quintics=True).keys())
if len(poly_solns) < deg:
result = ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes)
if gen != symbol:
y = Dummy('y')
lhs, rhs_s = invert_func(gen, y, symbol)
if lhs is symbol:
result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
else:
result = ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes)
else:
result = ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes)
if result is not None:
if isinstance(result, FiniteSet):
# this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
# - sqrt(2)*I/2. We are not expanding for solution with free
# variables because that makes the solution more complicated. For
# example expand_complex(a) returns re(a) + I*im(a)
if all([s.free_symbols == set() and not isinstance(s, RootOf)
for s in result]):
s = Dummy('s')
result = imageset(Lambda(s, expand_complex(s)), result)
return result
else:
return ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes)
