Question

Решите систему уравнений ​

Answer 1

$\left\{\begin{array}{l}\sqrt{a}+\sqrt{b}=10\\\sqrt[4]{a}+\sqrt[4]{b}=4\end{array}\right\ \ \left\{\begin{array}{l}\sqrt{a}=x^2\ ,\ \sqrt{b}=y^2\\\sqrt[4]{a}=x\geq 0\ \ ,\ \ \sqrt[4]{b}=y\geq 0\end{array}\right\ \ \left\{\begin{array}{l}x^2+y^2=10\\x+y=4\end{array}\right\\\\\\\left\{\begin{array}{l}x^2+y^2=10\\x^2+2xy+y^2=16\end{array}\right\ \ \left\{\begin{array}{l}x^2+y^2=10\\2xy+10=16\end{array}\right\ \ \left\{\begin{array}{l}x^2+y^2=10\\xy=3\end{array}\right$

$\left\{\begin{array}{l}x^2+\frac{9}{x^2}=10\\y=\frac{3}{x}\; ,\; x\ne 0\end{array}\right\ \ \left\{\begin{array}{l}x^4-10x^2+9=0\\y=\frac{3}{x}\; ,\; x\ne 0\end{array}\right\ \ \left\{\begin{array}{l}x^2=1\; ,\; x^2=9\\y=\frac{3}{x}\; ,\; x\ne 0\end{array}\right\\\\\\\left\{\begin{array}{l}x=-1\\y=-3\end{array}\right\ \ \left\{\begin{array}{l}x=1\\y=3\end{array}\right\ \ \left\{\begin{array}{l}x=-3\\y=-1\end{array}\right\ \ \left\{\begin{array}{l}x=3\\y=1\end{array}\right$

$\left\{\begin{array}{l}\sqrt[4]{a}=1\ ,\ \sqrt[4]{b}=3\\a=1\ \ ,\ \ b=81\end{array}\right\ \ \ \left\{\begin{array}{l}\sqrt[4]{a}=3\ ,\ \sqrt[4]{b}=1\\a=81\ ,\ b=1\end{array}\right$

$Otvet:\ \ (\; 1\, ;\, 81)\ ,\ (\, 81\, ;\, 1\, )\; .$