Question

Chứng minh: a²+a+1/ a²-a+1 >0

Answer 1

Ta có: \(\dfrac{a^2+a+1}{a^2-a+1}\)

\(=\dfrac{a^2+2\cdot a\cdot0,5+0,25+0,75}{a^2-2\cdot a\cdot0,5+0,25+0,75}\)

\(=\dfrac{\left(a+0,5\right)^2+0,75}{\left(a-0,5\right)^2+0,75}\)

Ta có: \(\left(a+0,5\right)^2+0,75\ge0,75>0\forall a\); Dấu "=" xảy ra \(\Leftrightarrow a=-0,5\)

lại có: \(\left(a-0,5\right)^2+0,75\ge0,75>0\forall a\); Dấu "=" xảy ra \(\Leftrightarrow a=0,5\)

do đó: \(\dfrac{\left(a+0,5\right)^2+0,75}{\left(a-0,5\right)^2+0,75}>0\forall a\); Dấu "=" xảy ra \(\Leftrightarrow a=0,5\)

hay \(\dfrac{a^2+a+1}{a^2-a+1}>0\)