Może ktoś powiedzieć dlaczego jest nie tak?
\(\displaystyle{ x^3+x-2=0\\
x=a+b\\
(a+b)^3+(a+b)-2=0\\
a^3+b^3+3a^2b+3ab^2 +(a+b)-2=0\\
a^3+b^3+(a+b)3ab +(a+b)-2=0\\
a^3+b^3+(a+b)(3ab+1)-2=0\\}\)
\(\displaystyle{ \begin{cases} a^3+b^3 = 2 \\ ab = - \frac{1}{3} \end{cases} \Rightarrow \begin{cases} a^3+b^3 = 2 \\ a^3b^3 = - \frac{1}{27} \end{cases}}\)
\(\displaystyle{ z^2-2z- \frac{1}{27}=0\\
\Delta = 4+ \frac{4}{27} = \frac{112}{27}\\\\
z = \frac{2 \pm \sqrt{ \frac{112}{27} } }{2}}\)
\(\displaystyle{ x= \sqrt[3]{\frac{2 + \sqrt{ \frac{112}{27} } }{2}} + \sqrt[3]{\frac{2 - \sqrt{ \frac{112}{27} } }{2}}}\)
Lecz ten wynik nie równa się jeden:
Kod: Zaznacz cały
http://www.wolframalpha.com/input/?i=%28%282%2B%28112%2F27%29^%281%2F2%29%29%2F2%29^%281%2F3%29%2B%28%282-%28112%2F27%29^%281%2F2%29%29%2F2%29%29^%281%2F3%29