Question

Решить систему уравнений: {√x *√y=10-3∜xy {2x-5y=6

Answer 1

$\left \{ {{\sqrt{x}\cdot \sqrt{y}=10-3\sqrt[4]{xy}} \atop {2x-5y=6}} \right. \; \; ,\; \; \; \; ODZ:\; x\geq 0\; ,\; y\geq 0\\\\a)\; \; \sqrt{xy}+3\sqrt[4]{xy}-10=0\\\\t=\sqrt[4]{xy} \geq 0\; ,\; \; \; t^2+3t-10=0\\\\t_1=-5,\; \; t_2=2\; \; (teorema\; Vieta)\\\\\sqrt[4]{xy}=-5\ \textless \ 0\; \; ne\; podxodit\\\\\sqrt[4]{xy}=2\; \; \Rightarrow \; \; \; xy=2^4\; ,\; \; xy=16$

$b)\; \; \left \{ {{xy=16} \atop {2x-5y=6}} \right. \; ,\; \left \{ {{\frac{5y+6}{2}\cdot y=16} \atop {x=\frac{5y+6}{2}}} \right. \; ,\; \left \{ {{5y^2+6y=32} \atop {x=\frac{5y+6}{2}} \right.$

$5y^2+6y-32=0\\\\D/4=9+160=169\; ,\\\\y_1= \frac{-3-13}{5}=-\frac{16}{5}=-3,2 \; \; ,\; \; y_2=\frac{-3+13}{5}=2\\\\x_1= \frac{5\cdot (-3,2)+6}{2} =-5\; ,\; \; x_2= \frac{5\cdot 2+6}{2}=8\\\\Otvet:\; \; (8;2)\; .$