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Question

cho 1/a+1/b+1/c=2 và a+b+c=abc . rút gọn N = 1/a2+1/b2+1/c2

Nguyễn Hoàng Minh · Accepted Answer

$a+b+c=abc\Leftrightarrow\dfrac{1}{ab}+\dfrac{1}{ac}+\dfrac{1}{bc}=1\Leftrightarrow\dfrac{2}{ab}+\dfrac{2}{bc}+\dfrac{2}{ca}=2$Mà $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=2\Leftrightarrow\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^2=4$$\Leftrightarrow\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2}{ab}+\dfrac{2}{bc}+\dfrac{2}{ac}=4\
\Leftrightarrow\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+2=4\
\Leftrightarrow\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=2$Câu trả lời: $a+b+c=abc\Leftrightarrow\dfrac{1}{ab}+\dfrac{1}{ac}+\dfrac{1}{bc}=1\Leftrightarrow\dfrac{2}{ab}+\dfrac{2}{bc}+\dfrac{2}{ca}=2$Mà $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=2\Leftrightarrow\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^2=4$$\Leftrightarrow\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2}{ab}+\dfrac{2}{bc}+\dfrac{2}{ac}=4\
\Leftrightarrow\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+2=4\
\Leftrightarrow\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=2$

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