※ 引述《sunfallmoon (落月)》之銘言：
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2 2
x , y ∈ Z , x + 4xy + 5y - 2x - 8y + 4 = 0 , 求 x - y 最小值
令所求為 t ,則 x = y + t .
2 2 2 2
代入原式 ( y + 2yt + t ) + ( 4y + 4yt ) + 5y - ( 2y + 2t ) - 8y + 4 = 0
2 2 2 2
→ t + ( 2y + 4y -2 ) * t + ( y + 4y + 5y - 2y - 8y + 4 ) = 0
2 2
→ t + ( 6y - 2 ) * t + ( 10y - 10y + 4 ) = 0
2 2
t 有最小值時,判別式 D = ( 6y -2 ) - 4 * ( 10y - 10y + 4 ) = 0
2 2
→ -4y + 16y - 12 = 0 → y - 4y + 3 = 0 → y = 1 or 3
2 2
(1) 若 y = 1 , x + 4x + 5 - 2x - 8 + 4 = 0 → x + 2x + 1 = 0 → x = -1
2 2
(2) 若 y = 3 , x + 12x + 45 - 2x -24 + 4 = 0 → x + 10x + 25 = 0 → x = -5
→ 當 x = -5 , y = 3 時, x - y 有最小值 -8