Zadatak 0235

Test: MPsIIJAII_048

 

U jednačini   \((k-1)x^2+(k-5)x-(k+2)=0\)   odredi k tako da je    \(\frac{1}{x_{1}}+\frac{1}{x_{2}}>2\)

 

Rešenje:


 

\((k-1)x^2+(k-5)x-(k+2)=0\)

 

\(x_{1}+x_{2}=-\frac{k-5}{k-1}\)

 

\(x_{1}\cdot x_{2}=-\frac{k+2}{k-1}\)

 

\(\frac{1}{x_{1}}+\frac{1}{x_{2}}>2\)  \(\Leftrightarrow \frac{x_{1}+x_{2}}{x_{1}\cdot x_{2}}>2\) \(\Leftrightarrow \frac{-\frac{k-5}{k-1}}{-\frac{k+2}{k-1}}>2\)

 

\(\Leftrightarrow \frac{k-5}{k+2}-2>0\)  \(\Leftrightarrow \frac{-k-9}{k+2}>0\)

 

slika uz zadatak 0235

 

\(k\in (-9,-2)\).