Question

Решить систему
X^2+y^2=37
Xy=6

Answer 1

$left { {{x^2+y^2=37} atop {xy=6}} right. left { {{x^2+y^2=37} atop {y= frac{6}{x} }} right. left { {{x^2+( frac{6}{x})^2=37} atop {y= frac{6}{x} }} right. left { {{x^4-37x^2+36=0} atop {y= frac{6}{x}}} right. \ \ t=x^2,t geq 0 \ \ t^2-37t+36=0;D=1369-144=1225=35^2 \ \ 1) t_{1} =(37+35)/2=36=6^2; left { {{ x_{1}=6 } atop { y_{1} =1}} right. ; left { {{ x_{2}=-6 } atop { y_{2} =-1}} right.$
$2) t_{2} =(37-35)/2=1; left { {{ x_{3}=1 } atop { y_{3} =6}} right. ; left { {{ x_{4}=-1 } atop { y_{4} =-6}} right. \ \ otvet:(6;1)(-6;-1)(1;6)(-1;-6)$



$left { {{x^2+y^2=37} atop {xy=6}} right. left { {{x^2+2xy+y^2-2xy=37} atop {xy=6}} right. left { {{(x+y)^2-2xy=37} atop {xy=6}} right. left { {{(x+y)^2-12=37} atop {xy=6}} right. \ \ left { {{(x+y)^2-49=0} atop {xy=6}} right. left { {{(x+y+7)(x+y-7)=0} atop {xy=6}} right. \ \ 1)left { {{x+y+7=0} atop {xy=6}} right. left { {{y=-x-7} atop {xy=6}} right. left { {{y=-x-7} atop {x(-x-7)=6}} right.left { {{y=-x-7} atop {x^2+7x+6=0}} right.$
D=49-24=25
$left { {{ x_{1}=(-7-5)/2=-6 } atop { y_{1}=6-7=-1 }} right. left { {{ x_{2}=(-7+5)/2=-1 } atop { y_{1}=1-7=-6 }} right. \ \ 2)left { {{x+y-7=0} atop {xy=6}} right. left { {{y=7-x} atop {xy=6}} right. left { {{y=7-x} atop {x(7-x)=6}} right.left { {{y=7-x} atop {x^2-7x+6=0}} right. \ D=25 \ \ left { {{ x_{3}=(7-5)/2=1 } atop { y_{3}=7-1=6 }} right. left { {{ x_{4}=(7+5)/2=6} atop { y_{4}=7-6=1 }} right.$