已知X+Y=3,XY=-1,求2(X-Y)^4-9(X-Y)^2+12XY

已知X+Y=3,XY=-1,求2(X-Y)^4-9(X-Y)^2+12XY
已知X+Y=3,XY=-1,求2(X-Y)^4-9(X-Y)^2+12XY

已知X+Y=3,XY=-1,求2(X-Y)^4-9(X-Y)^2+12XY
X+Y=3
(x+y)^2=9
(x-y)^2+4xy=9
(x-y)^2=9-4(-1)=13
2(X-Y)^4-9(X-Y)^2+12XY
=2*13^2-9*13+12(-1)
=338-117-12
=209

(X-Y)²=(X+Y)²-4XY=3²+4=13
则(X-Y)^4=[(X-Y)²]²=13²=169
所以
2(X-Y)^4-9(X-Y)^2+12XY
=2×169-9×13-12
=209

(x-y)^2=(x+y)^2-4xy=9+4=13
所以原式=2*169-9*13-12=209

209