若a.b.c三數皆相異，且滿足下列等式：
a^3+b^3+2(a^2+b^2)=b^3+c^3+2(b^2+c^2)=c^3+a^3+2(c^2+a^2)
則a+b+c=＿＿＿
資優數學 幫幫我
2009-06-24 2:52 am
回答 (3)
2009-06-24 3:09 am
✔ 最佳答案
若a.b.c三數皆相異，且滿足下列等式：a^3+b^3+2(a^2+b^2)=b^3+c^3+2(b^2+c^2)=c^3+a^3+2(c^2+a^2)
則a+b+c=＿＿＿
Sol
2P= a^3+b^3+2(a^2+b^2)=b^3+c^3+2(b^2+c^2)=c^3+a^3+2(c^2+a^2)
6P=2a^3+2b^3+2c^3+4(a^2+b^2+c^2)
3P=a^3+b^3+c^3+2(a^2+b^2+c^2)
P=a^3+2a^2=b^3+2b^2=c^3+2c^2
So a,b,c 為 x^3+2x^2－p=0 之三根
So a+b+c=－2
2009-06-24 8:55 am
a^3 + b^3 + 2(a^2 + b^2) = b^3 + c^3 + 2(b^2 + c^2)
a^3 + 2a^2 = c^3 + 2c^2
a^3 - c^3 + 2(a^2 - c^2) = 0
a^2 + ac + c^2 + 2(a + c) = 0
同理得
b^2 + bc + c^2 + 2(b + c) = 0
相減得
a^2 - b^2 + ac - bc + 2(a - b) = 0
a + b + c = -2
a^3 + 2a^2 = c^3 + 2c^2
a^3 - c^3 + 2(a^2 - c^2) = 0
a^2 + ac + c^2 + 2(a + c) = 0
同理得
b^2 + bc + c^2 + 2(b + c) = 0
相減得
a^2 - b^2 + ac - bc + 2(a - b) = 0
a + b + c = -2
2009-06-24 3:20 am
漂亮解法!
收錄日期: 2021-05-04 00:44:34
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