Let a,b and c are raional numbers which satisfy a-7b+8c=4 and 8a+4b-c=7Then2 2 2 a -b +c =________2 2 2 a -b +c 表示: a平方-b平方+c平方

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Let a,b and c are raional numbers which satisfy a-7b+8c=4 and 8a+4b-c=7Then2 2 2 a -b +c =________2 2 2 a -b +c 表示: a平方-b平方+c平方
Let a,b and c are raional numbers which satisfy
a-7b+8c=4 and 8a+4b-c=7
Then
2 2 2
a -b +c =________
2    2   2          
a  -b +c      表示: a平方-b平方+c平方 

Let a,b and c are raional numbers which satisfy a-7b+8c=4 and 8a+4b-c=7Then2 2 2 a -b +c =________2 2 2 a -b +c 表示: a平方-b平方+c平方
用a^2表示a的平方了.
因为a-7b+8c=4,所以 a+8c=7b+4 (1)
同理由 8a+4b-c=7 可知 8a-c=7-4b (2)
(1)(2)两式平方相加得到:
(a+8c)^2+(8a-c)^2=(7b+4)^2+(7-4b)^2 (3)
利用和的平方公式将上式中的平方项展开得到:
(a+8c)^2=a^2+16ac+64c^2;
(8a-c)^2=64a^2-16ac+c^2;
所以等式左端=65(a^2+c^2)
同理(7b+4)^2=49b^2+56b+16;(7-4b)^2=49-56b+16b^2,所以
等式右端=65(b^2+1).
这样,将上述结果代入(3)式得到:
65(a^2+c^2)=65(b^2+1).
两边约去65并移项即得:a^2-b^2+c^2=1.

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