Question

8x²y² + x² + y²=10xy

tìm các số nguyên x,y thỏa mãn

 

Answer 1

8x2y2+x2+y2=10xy8x2y2+x2+y2=10xy

⇔8x2y2−8xy+x2+y2−2xy=0⇔8x2y2-8xy+x2+y2-2xy=0

⇔2(4x2y2−4xy+1)+x2+y2−2xy=2⇔2(4x2y2-4xy+1)+x2+y2-2xy=2

⇔2(2xy−1)2+(x−y)2=2⇔2(2xy-1)2+(x-y)2=2

Nếu(2xy−1)2=0⇒(x−y)2=2(2xy-1)2=0⇒(x-y)2=2(vô nghiệm)

Nếu2(2xy−1)2=2⇒(x−y)2=0⇒x=y2(2xy-1)2=2⇒(x-y)2=0⇒x=y

(2x2−1)2=1⇒(2x2-1)2=1⇒[2x2−1=√12x2−1=√−1[2x2−1=12x2−1=−1 ⇒[x=−1;1x=0[x=−1;1x=0 

Nếu(2xy−1)2≥2⇒2=2(2xy−1)2+(x−y)2≥4(2xy-1)2≥2⇒2=2(2xy-1)2+(x-y)2≥4(vô nghiệm)

Vậy (x;y)(x;y) thỏa mãn các cặp là (0;0);(1;1);(−1;−1)(0;0);(1;1);(-1;-1)