Question

cho a+b+c=0tinh B=ab/a²+b²-c²+bc/b²+c²-a²+ca/c²+a²-b²

Answer 1

Với a + b + c = 0 ta có:

\(B=\dfrac{ab}{a^2+b^2-c^2}+\dfrac{bc}{b^2+c^2-a^2}+\dfrac{ca}{c^2+a^2-b^2}\)

\(\Leftrightarrow B=\dfrac{ab}{\left(a+b\right)^2-2ab-c^2}+\dfrac{bc}{\left(b+c\right)^2-2bc-a^2}+\dfrac{ca}{\left(c+a\right)^2-2ca-b^2}\)

\(\Leftrightarrow B=\dfrac{ab}{\left(a+b+c\right)\left(a+b-c\right)-2ab}+\dfrac{bc}{\left(b+c-a\right)\left(b+c+a\right)-2bc}+\dfrac{ac}{\left(a+c+b\right)\left(c+a-b\right)-2ca}\)

\(\Leftrightarrow B=\dfrac{ab}{-2ab}+\dfrac{bc}{-2bc}+\dfrac{ac}{-2ac}\)

\(\Leftrightarrow B=\dfrac{-1}{2}+\dfrac{-1}{2}+\dfrac{-1}{2}\)

\(\Leftrightarrow B=\dfrac{-3}{2}\)