Question

مطلوب است محاسبه xوyبه شرطی که: $x^{2} +xy+2= y^{2} +y$و$( x^{2}+2)(x-y+1)=-3y$ دسته مطلوب عبارت است از: $(i \sqrt{2},0);(-i \sqrt{2} ,0);(0,-2);(-1,-3);(-1,1);(4,6)$

Answer 1

$$48-8x- x^{2} =48-2x+9-9-6x- x^{2} =2(24-x)+9- (x+3)^{2};;x+3=a,x-24=b \Longrightarrow (x+3)^{2} (48-8x- x^{2} )= (x-24)^{2}= a^{2} (-2b+9- a^{2} )= b^{2} \Longrightarrow a^{4} +2 a^{2} b+ b^{2} =9 a^{2} \Longrightarrow ( a^{2}+b )^{2}=9 a^{2} \Longrightarrow a^{2} +b=3a ;or; a^{2} +b=-3a \Longrightarrow (x+3)^{2}+x-24=3x+9 \Longrightarrow x^{2}+4x-24=0 \Longrightarrow x=-2+2 \sqrt{7} ;or;x=-2-2 \sqrt{7} ,... \Longrightarrow S= -2-2 \sqrt{7};or;-5- \sqrt{31} in domain$$