Toán Lớp 9: Tìm tất cả cặp số nguyên `(x;y)` thoả mãn $x^{2}$ `+` `5xy` `+` `6`$y^{2}$ `+` `x` `+` `2y` `-` `2` `=` `0`

Question

Toán Lớp 9:  Tìm tất cả cặp số nguyên (x;y) thoả mãn    $x^{2}$ + 5xy + 6$y^{2}$ + x + 2y - 2 = 0

cattien · Answer

Bài làm:        $x^2+5xy+6y^2+x+2y-2=0$    &hArr; $(x^2+2xy)+(3xy+6y^2)+(x+2y)-2=0$    &hArr; $x(x+2y)+3y(x+2y)+(x+2y)=2$    &hArr; $(x+2y)(x+3y)+(x+2y)=2$    &hArr; $(x+2y)(x+3y+1)=2$    Vì x, y &isin; Z &rArr; $x+2y &isin; Z$ và $x+3y+1 &isin; Z$   &rArr; $x+2y$ và $x+3y+1$ cùng thu&ocirc;̣c ước của 2   &rArr; $(x+2y)(x+3y+1)=1.2=2.1=(-1).(-2)=(-2).(-1)$   +) TH1: $ left  { {{x+2y=1}  atop {x+3y+1=2}}  right.$ &hArr; $ left  { {{x+2y=1}  atop {x+3y=1}}  right.$      &hArr; $ left  { {{y=0}  atop {x+2y=1}}  right.$ &hArr; $ left  { {{y=0}  atop {x=1}}  right.$ ( thỏa mãn)$(1;0)$   +) TH2: $ left  { {{x+2y=2}  atop {x+3y+1=1}}  right.$ &hArr; $ left  { {{x+2y=2}  atop {x+3y=0}}  right.$      &hArr; $ left  { {{y=-2}  atop {x+2y=2}}  right.$ &hArr; $ left  { {{y=-2}  atop {x=2-2y}}  right.$      &hArr; $ left  { {{y=-2}  atop {x=6}}  right.$ ( thỏa mãn )   +) TH3: $ left  { {{x+2y=-1}  atop {x+3y+1=-2}}  right.$ &hArr; $ left  { {{x+2y=-1}  atop {x+3y=-3}}  right.$      &hArr; $ left  { {{y=-2}  atop {x+2y=-1}}  right.$ &hArr; $ left  { {{y=-2}  atop {x=-1-2y}}  right.$      &hArr; $ left  { {{y=-2}  atop {x=3}}  right.$ ( thỏa mãn )   +) TH4: $ left  { {{x+2y=-2}  atop {x+3y+1=-1}}  right.$ &hArr; $ left  { {{x+2y=-2}  atop {x+3y=-2}}  right.$      &hArr; $ left  { {{y=0}  atop {x+2y=-2}}  right.$ &hArr; $ left  { {{y=0}  atop {x=-2}}  right.$ ( thỏa mãn )   V&acirc;̣y các cặp nghi&ecirc;̣m $(x;y)$ thỏa mãn đ&ecirc;̀ bài là $(1;0)$ ; $(6;-2)$ ; $(3;-2)$ ; $(-2;0)$