Izvorna datoteka (SVG datoteka, nominalno 14.406 × 1.443 piksela, veličina datoteke: 326 KB)
This image shows some kind of formula that could be converted to
TeX . Storing formulas as images makes it harder to change them. TeX also helps making sure that they all use the same font and size.
A replacement has been proposed:
r
1
=
−
a
4
−
1
2
a
2
4
−
2
b
3
+
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
−
1
2
a
2
2
−
4
b
3
−
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
−
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
−
−
a
3
+
4
a
b
−
8
c
4
a
2
4
−
2
b
3
+
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
r
2
=
−
a
4
−
1
2
a
2
4
+
2
b
3
+
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
−
1
2
a
2
2
−
4
b
3
−
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
−
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
−
−
a
3
+
4
a
b
−
8
c
4
a
2
4
−
2
b
3
+
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
r
3
=
−
a
4
+
1
2
a
2
4
−
2
b
3
+
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
−
1
2
a
2
2
−
4
b
3
−
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
−
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
−
−
a
3
+
4
a
b
−
8
c
4
a
2
4
−
2
b
3
+
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
r
4
=
−
a
4
+
1
2
a
2
4
+
2
b
3
+
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
−
1
2
a
2
2
−
4
b
3
−
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
−
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
−
−
a
3
+
4
a
b
−
8
c
4
a
2
4
−
2
b
3
+
2
1
3
(
b
2
−
3
a
c
+
12
d
)
3
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
)
1
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
+
−
4
(
b
2
−
3
a
c
+
12
d
)
3
+
(
2
b
3
−
9
a
b
c
+
27
c
2
+
27
a
2
d
−
72
b
d
)
2
54
)
1
3
{\displaystyle {\begin{aligned}r_{1}&={\frac {-a}{4}}-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}\\&-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{2}}-{\frac {4b}{3}}-{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}-\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}-{\frac {-a^{3}+4ab-8c}{4{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}}}}}\\r_{2}&={\frac {-a}{4}}-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{4}}+{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}\\&-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{2}}-{\frac {4b}{3}}-{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}-\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}-{\frac {-a^{3}+4ab-8c}{4{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}}}}}\\r_{3}&={\frac {-a}{4}}+{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}\\&-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{2}}-{\frac {4b}{3}}-{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}-\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}-{\frac {-a^{3}+4ab-8c}{4{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}}}}}\\r_{4}&={\frac {-a}{4}}+{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{4}}+{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}\\&-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{2}}-{\frac {4b}{3}}-{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}-\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}-{\frac {-a^{3}+4ab-8c}{4{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}}}}}\end{aligned}}}
In your article, replace the image with:
<math>\begin{align}
r_1 & =\frac{-a}{4}-\frac{1}{2}{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } \\
& -\frac{1}{2}{\sqrt{\frac{a^{2} }{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} }-\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } } } } \\
r_2 & =\frac{-a}{4}-\frac{1}{2}{\sqrt{\frac{a^{2} }{4}+\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } \\
& -\frac{1}{2}{\sqrt{\frac{a^{2} }{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} }-\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } } } } \\
r_3 & =\frac{-a}{4}+\frac{1}{2}{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } \\
& -\frac{1}{2}{\sqrt{\frac{a^{2} }{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} }-\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } } } } \\
r_4 & =\frac{-a}{4}+\frac{1}{2}{\sqrt{\frac{a^{2} }{4}+\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } \\
& -\frac{1}{2}{\sqrt{\frac{a^{2} }{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} }-\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } } } }
\end{align}</math> Deutsch ∙ English ∙ italiano ∙ magyar ∙ Nederlands ∙ polski ∙ sicilianu ∙ svenska ∙ Ελληνικά ∙ български ∙ македонски ∙ русский ∙ 日本語 ∙ فارسی ∙ +/−
Sažetak Licenciranje
Public domain Public domain false false
Ova slika je javno dobro („public domain “), jer sadrži samo opće informacije koje ne dostižu kreativni nivo potreban za copyright.
bosanski Add a one-line explanation of what this file represents
engleski All solutions of the equation $x^4+ax^3+bx^2+cx+d=0$
Historija datoteke
Kliknite na datum/vrijeme da vidite verziju datoteke iz tog vremena.
Datum/vrijeme Smanjeni pregled Dimenzije Korisnik Komentar trenutno 02:10, 17 maj 2013 14.406 × 1.443 (326 KB) Linket {{subst:Upload marker added by en.wp UW}} {{Information |Description = {{en|All 4 roots of a quartic equation (x^4+ax^3+bx^2+cx+d=0).}} |Source = http://planetmath.org/quarticformula |Author = David Jao }} Category:Mathematical equations
Upotreba datoteke
Sljedeća stranica koristi ovu datoteku:
Globalna upotreba datoteke
Sljedeći wikiji koriste ovu datoteku:
Upotreba na bn.wikipedia.org
Upotreba na de.wikipedia.org
Upotreba na en.wikipedia.org
Upotreba na fi.wikipedia.org
Upotreba na sh.wikipedia.org
Ova datoteka sadržava dodatne podatke koje je vjerovatno dodala digitalna kamera ili skener u procesu snimanja, odnosno digitalizacije. Ako je datoteka mijenjana, podaci možda nisu u skladu sa stvarnim stanjem.
