三次方程式解くだけ(2x^3+x^2-x+9=9)

$2x^3+x^2-x+9=0$

$x^3+\dfrac{1}{2}x^2-\dfrac{1}{2}x+\dfrac{9}{2}=0$

$x=y-\dfrac{1}{6}$

$x^3+\dfrac{1}{2}x^2-\dfrac{1}{2}x+\dfrac{9}{2}$

$=(y-\dfrac{1}{6})^3+\dfrac{1}{2}(y-\dfrac{1}{6})^2-\dfrac{1}{2}(y-\dfrac{1}{6})+\dfrac{9}{2}$

$=a+\dfrac{1}{2}b-\dfrac{1}{2}(y-\dfrac{1}{6})+\dfrac{9}{2}$

$a=(y-\dfrac{1}{6})^3=y^3-\dfrac{1}{2}y^2+\dfrac{1}{12}y-\dfrac{1}{216}$

$b=(y-\dfrac{1}{6})^2=y^2-\dfrac{1}{3}y+\dfrac{1}{36}$

$y^3-\dfrac{7}{12}y+\dfrac{124}{27}$

$y=v+w$

$(v+w)^3-\dfrac{7}{12}(v+w)+\dfrac{124}{27}$

$=v^3+3v^2w+3vw^2-\dfrac{7}{12}v+w^3-\dfrac{7}{12}w+\dfrac{124}{27}$

$=(v^3+w^3+\dfrac{124}{27})+3v^2w+3vw^2-\dfrac{7}{12}v-\dfrac{7}{12}w$

$=(v^3+w^3+\dfrac{124}{27})+(v+w)(3vw-\dfrac{7}{12})=0$

$\begin{cases}v^3+w^3+\dfrac{124}{27}=0 \\ 3vw-\dfrac{7}{12}=0 \end{cases}$

$w=\dfrac{7}{36v}$

$v^3+(\dfrac{7}{36v})^3+\dfrac{124}{27}$

$v^3+\dfrac{343}{46656v^3}+\dfrac{124}{27}=0$

$v^6+\dfrac{343}{46656}+\dfrac{124}{27}v^3=0$

$v^3=\dfrac{-\dfrac{124}{27}\pm\sqrt{(\dfrac{124}{27})^2-4\cdot\dfrac{343}{46656}}}{2}$

$\dfrac{\sqrt{337}}{8}-\dfrac{62}{27}$

$v=\sqrt[3]{\dfrac{\sqrt{337}}{8}-\dfrac{62}{27}},\sqrt[3]{\dfrac{\sqrt{337}}{8}-\dfrac{62}{27}}\omega,\sqrt[3]{\dfrac{\sqrt{337}}{8}-\dfrac{62}{27}}\omega^2$

$w=\sqrt[3]{\dfrac{\sqrt{337}}{8}+\dfrac{62}{27}},\sqrt[3]{\dfrac{\sqrt{337}}{8}+\dfrac{62}{27}}\omega,\sqrt[3]{\dfrac{\sqrt{337}}{8}+\dfrac{62}{27}}\omega^2$

$y=\sqrt[3]{\dfrac{\sqrt{337}}{8}+\dfrac{62}{27}}+\sqrt[3]{\dfrac{\sqrt{337}}{8}-\dfrac{62}{27}},$

$\sqrt[3]{\dfrac{\sqrt{337}}{8}+\dfrac{62}{27}}\omega^2+\sqrt[3]{\dfrac{\sqrt{337}}{8}-\dfrac{62}{27}}\omega$

$\sqrt[3]{\dfrac{\sqrt{337}}{8}+\dfrac{62}{27}}\omega+\sqrt[3]{\dfrac{\sqrt{337}}{8}-\dfrac{62}{27}}\omega^2$

$x=\sqrt[3]{\dfrac{\sqrt{337}}{8}+\dfrac{62}{27}}+\sqrt[3]{\dfrac{\sqrt{337}}{8}-\dfrac{62}{27}}-\dfrac{1}{6},$

$\sqrt[3]{\dfrac{\sqrt{337}}{8}+\dfrac{62}{27}}\omega^2+\sqrt[3]{\dfrac{\sqrt{337}}{8}-\dfrac{62}{27}}\omega-\dfrac{1}{6}$

$\sqrt[3]{\dfrac{\sqrt{337}}{8}+\dfrac{62}{27}}\omega+\sqrt[3]{\dfrac{\sqrt{337}}{8}-\dfrac{62}{27}}\omega^2-\dfrac{1}{6}$

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