已知 x (x^2 - 3 y^2) = √421, y (y^2 - 3 x^2) = 40，求 x^2+y^2

uk702

已知
\(\begin{align} x(x^2 - 3y^2) &= \sqrt{ 421}\\y(y^2 - 3x^2) &= 40 \end{align}\)
求 \( x^2+y^2 \)。

不知是否有那种软件，不仅能给出公式解，还能进行化简。

hujunhua

⑴-i⑵得\[(x+iy)^3=\sqrt{421}-40i\tag{3}\]
取模可得\[x^2+y^2=\sqrt[3]{2021}\]原来是一个2021年份题。
知道答案就好做了，直接`\D ⑴^2+⑵^2`得\[(x^2+y^2)^3=2021\tag{4}\]

mathematica

hujunhua 发表于 2021-1-11 07:21
⑴-i⑵得\[(x+iy)^3=\sqrt{421}-40i\]
取模可得\[x^2+y^2=\sqrt[3]{2021}\]原来是一个2021年份题。
知道 ...

漏解了！
不过漏解很正常！
因为你没软件计算！

1. Clear["Global`*"];
   
2. out=Solve[{x*(x^2-3*y^2)==Sqrt[421],y*(y^2-3*x^2)==40,x^2+y^2==u},{x,y,u}];
   
3. aa=FullSimplify[u/.out]
   
4. bb=Union[aa]
   
5. cc=N@bb
   

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运行结果如下：
\[\left\{\sqrt[3]{2021},\text{Root}\left[\text{$\#$1}^3-2021\&,3\right],-\sqrt[3]{-2021},\text{Root}\left[\text{$\#$1}^3-2021\&,3\right],\sqrt[3]{2021},-\sqrt[3]{-2021},\text{Root}\left[\text{$\#$1}^3-2021\&,3\right],\sqrt[3]{2021},-\sqrt[3]{-2021}\right\}\]
\[\left\{-\sqrt[3]{-2021},\sqrt[3]{2021},\text{Root}\left[\text{$\#$1}^3-2021\&,3\right]\right\}\]
\[\{-6.32158-10.9493 i,12.6432,-6.32158+10.9493 i\}\]

TSC999

1. NSolve[{x*(x^2 - 3*y^2) == Sqrt[421], y*(y^2 - 3*x^2) == 40,
   
2.   x^2 + y^2 == u}, {u, x, y}, Reals]

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运行结果如下：
{{u->12.6431, x->-2.76128, y->-2.24019},{u->12.6431, x->-0.55942, y->3.51143},{u->12.6431, x->3.32070, y->-1.27124}}

mathematica

TSC999 发表于 2021-1-11 20:51
运行结果如下：
{{u->12.6431, x->-2.76128, y->-2.24019},{u->12.6431, x->-0.55942, y->3.51143},{u-> ...

啥时候假设x y是实数了？

mathematica

TSC999 发表于 2021-1-11 20:51
运行结果如下：
{{u->12.6431, x->-2.76128, y->-2.24019},{u->12.6431, x->-0.55942, y->3.51143},{u-> ...

1. Clear["Global`*"];
   
2. out=Grid@NSolve[{x*(x^2-3*y^2)==Sqrt[421],y*(y^2-3*x^2)==40,x^2+y^2==u},{x,y,u}]
   

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\[\begin{array}{ccc}
x\to 1.38064\, -2.39134 i & y\to 1.1201\, -1.94006 i & u\to -6.32158-10.9493 i \\
x\to 1.38064\, +2.39134 i & y\to 1.1201\, +1.94006 i & u\to -6.32158+10.9493 i \\
x\to -2.76128 & y\to -2.24019 & u\to 12.6432 \\
x\to 3.32071 & y\to -1.27125 & u\to 12.6432 \\
x\to -1.66035-2.87582 i & y\to 0.635623\, +1.10093 i & u\to -6.32158+10.9493 i \\
x\to -1.66035+2.87582 i & y\to 0.635623\, -1.10093 i & u\to -6.32158-10.9493 i \\
x\to 0.279711\, +0.484474 i & y\to -1.75572-3.041 i & u\to -6.32158+10.9493 i \\
x\to 0.279711\, -0.484474 i & y\to -1.75572+3.041 i & u\to -6.32158-10.9493 i \\
x\to -0.559422 & y\to 3.51144 & u\to 12.6432 \\
\end{array}\]