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\(B=\dfrac{\left(4a^2-1\right)\left(b-c\right)-\left(4b^2-1\right)\left(a-c\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}+\dfrac{4c^2-1}{\left(a-c\right)\left(b-c\right)}\)
\(=\dfrac{4a^2b-4a^2c-b+c-4ab^2+4b^2c+a-c}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}+\dfrac{4ac^2-4bc^2-a+b}{\left(a-c\right)\left(b-c\right)\left(a-b\right)}\)
\(=\dfrac{4a^2b-4a^2c+a-b-4ab^2+4b^2c+4ac^2-4bc^2-a+b}{\left(a-c\right)\left(b-c\right)\left(a-b\right)}\)
\(=\dfrac{4a^2b-4ab^2-4a^2c+4ac^2-4bc^2+4b^2c}{\left(a-c\right)\left(b-c\right)\left(a-b\right)}\)
\(=\dfrac{4a^2\left(b-c\right)+4bc\left(b-c\right)-4a\left(b^2-c^2\right)}{\left(b-c\right)\left(a-c\right)\left(a-b\right)}\)
\(=\dfrac{4a^2+4bc-4a\left(b+c\right)}{\left(a-c\right)\left(a-b\right)}\)
\(=\dfrac{4a^2-4ab+4bc-4ac}{\left(a-c\right)\left(a-b\right)}\)
\(=\dfrac{4a\left(a-b\right)-4c\left(a-b\right)}{\left(a-c\right)\left(a-b\right)}=4\)
Tìm trước khi hỏi Câu hỏi của Phan Đình Trường - Toán lớp 8 | Học trực tuyến
\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{a+b+c}\)
\(\Leftrightarrow\left(\dfrac{1}{a}+\dfrac{1}{b}\right)+\left(\dfrac{1}{c}-\dfrac{1}{a+b+c}\right)=0\)
\(\Leftrightarrow\dfrac{a+b}{ab}+\dfrac{a+b+c-c}{c\left(a+b+c\right)}=0\)
\(\Leftrightarrow\dfrac{a+b}{ab}+\dfrac{a+b}{c\left(a+b+c\right)}=0\)
\(\Leftrightarrow\left(a+b\right)\times\dfrac{ac+bc+c^2+ab}{abc\left(a+b+c\right)}=0\)
\(\Leftrightarrow\dfrac{\left(a+b\right)\left(a+c\right)\left(b+c\right)}{abc\left(a+b+c\right)}=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a=-b\\b=-c\\c=-a\end{matrix}\right.\)
\(\Rightarrow N=0\)
\(\dfrac{a}{\left(a-b\right)\left(a-c\right)}+\dfrac{b}{\left(b-c\right)\left(b-a\right)}+\dfrac{c}{\left(c-a\right)\left(c-b\right)}=\dfrac{a}{\left(a-b\right)\left(a-c\right)}-\dfrac{b}{\left(b-c\right)\left(a-b\right)}+\dfrac{c}{\left(a-c\right)\left(b-c\right)}=\dfrac{a\left(b-c\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}-\dfrac{b\left(a-c\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}+\dfrac{c\left(a-b\right)}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}=\dfrac{ab-ac}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}-\dfrac{ab-bc}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}+\dfrac{ac-bc}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}=\dfrac{ab-ac-ab+bc+ac-bc}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}=\dfrac{0}{\left(a-b\right)\left(a-c\right)\left(b-c\right)}=0\)
Đề sai rồi: a,b,c > 0 thì làm sao mà có: ab + bc + ca = 0 được.
điện thoại cùi nên chụp hơi mờ, đề này còn thiếu a,,bc>0
\(N=\dfrac{\left(a-b\right)\left(b+c\right)\left(a+c\right)+\left(b-c\right)\left(a+b\right)\left(c+a\right)+\left(c-a\right)\left(a+b\right)\left(b+c\right)+\left(a-b\right)\left(b-c\right)\left(c-a\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)\(=\dfrac{\left(a+c\right)\left(ab-b^2+ac-bc+ab-ac+b^2-cb\right)+\left(c-a\right)\left(ab+b^2+ac+bc+ab-b^2-ac+cb\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)\(=\dfrac{\left(a+c\right)\left(2ab-2bc\right)+\left(c-a\right)\left(2ab+2bc\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)
\(=\dfrac{2b\left(a+c\right)\left(a-c\right)+2b\left(c-a\right)\left(a+c\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}=\dfrac{2b\left(c+a\right)\left(a-c+c-a\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}=0\)
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