已知x,y为正整数,并且x*y+x+y=71,x*x*y+x*y*y=880,求3*x*x+8*x*y+3*y*y的值
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已知x,y为正整数,并且x*y+x+y=71,x*x*y+x*y*y=880,求3*x*x+8*x*y+3*y*y的值
![已知x,y为正整数,并且x*y+x+y=71,x*x*y+x*y*y=880,求3*x*x+8*x*y+3*y*y的值](/uploads/image/z/18341296-16-6.jpg?t=%E5%B7%B2%E7%9F%A5x%2Cy%E4%B8%BA%E6%AD%A3%E6%95%B4%E6%95%B0%2C%E5%B9%B6%E4%B8%94x%2Ay%2Bx%2By%3D71%2Cx%2Ax%2Ay%2Bx%2Ay%2Ay%3D880%2C%E6%B1%823%2Ax%2Ax%2B8%2Ax%2Ay%2B3%2Ay%2Ay%E7%9A%84%E5%80%BC)
xy+(x+y)=71
xy(x+y)=880
令xy=A,x+y=B,即:
A+B=71
AB=880
解之得:
A=16,B=55 或 A=55,B=16.
当A=16,B=55时:
x = (3*329^(1/2))/2 + 55/2 = 54.70753572082558159734942755459 > 0
y = 55/2 - (3*329^(1/2))/2 = 0.29246427917441840265057244541003 > 0
或
x = 55/2 - (3*329^(1/2))/2 = 0.29246427917441840265057244541003 > 0
y = (3*329^(1/2))/2 + 55/2 = 54.70753572082558159734942755459 > 0
当A=55,B=16时:
x = 11 > 0
y = 5 > 0
或
x = 5 > 0
y = 11 > 0
所以两组情况都成立.
3x²+8xy+3y²
=3(x²+y²)+8xy
=3(x+y)²+2xy
=3B²+2A
所以当A=16,B=55时:原式=3*55²+32=9107
当A=55,B=16时,原式=3*16²+110=878
所以原式的值为:9107或878
xy(x+y)=880
令xy=A,x+y=B,即:
A+B=71
AB=880
解之得:
A=16,B=55 或 A=55,B=16.
当A=16,B=55时:
x = (3*329^(1/2))/2 + 55/2 = 54.70753572082558159734942755459 > 0
y = 55/2 - (3*329^(1/2))/2 = 0.29246427917441840265057244541003 > 0
或
x = 55/2 - (3*329^(1/2))/2 = 0.29246427917441840265057244541003 > 0
y = (3*329^(1/2))/2 + 55/2 = 54.70753572082558159734942755459 > 0
当A=55,B=16时:
x = 11 > 0
y = 5 > 0
或
x = 5 > 0
y = 11 > 0
所以两组情况都成立.
3x²+8xy+3y²
=3(x²+y²)+8xy
=3(x+y)²+2xy
=3B²+2A
所以当A=16,B=55时:原式=3*55²+32=9107
当A=55,B=16时,原式=3*16²+110=878
所以原式的值为:9107或878
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