1 reply to this topic

[Katyusha]

Giải PT $x=\sqrt{2-x}\sqrt{3-x}+\sqrt{3-x}\sqrt{5-x}+\sqrt{5-x}\sqrt{2-x}$

[ducvipdh12]

đặt $a=\sqrt{2-x};b=\sqrt{3-x},c=\sqrt{5-x}$ Từ giả thiết ta có:

$\left\{\begin{matrix} x=ab+bc+ca & & & \\ 2-a^2=x & & & \\ 3-b^2=x & & & \\ 5-c^2=x & & & \end{matrix}\right.$

$\Rightarrow \left\{\begin{matrix} (a+b)(a+c)=2 & & \\ (b+c)(b+a)=3 & & \\ (c+a)(c+b)=5 & & \end{matrix}\right.$

$\left\{\begin{matrix} (a+b)(b+c)(c+a)=\sqrt{30} & & & \\ (a+b)(a+c)=2 & & & \\ (b+c)(b+a)=3 & & & \\ (c+a)(c+b)=5 & & & \end{matrix}\right.$

....