Question

Cho a, b, c là ba số thực khác 0 thỏa mãn các điều kiện: a+b+c=0 và 1/a+1/b+1/c=3 tinh (1+1/a)^2+(1+1/b)^2+(1+1/c)^2

Answer 1

\(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2=9\Rightarrow\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)=9\)

\(\Rightarrow\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2\left(\frac{a+b+c}{abc}\right)=9\)

\(\Rightarrow\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=9\)

\(\left(1+\frac{1}{a}\right)^2+\left(1+\frac{1}{b}\right)^2+\left(1+\frac{1}{c}\right)^2=3+2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)=3+2.3+9=?\)