Pismeni zadaci

Zadatak 0232

Test: MPsIIJAII_048

 

Reši jednačinu:

 

\(x^4-\left ( a^2 +\frac{1}{a^2}\right )x^2+1=0\).

 

Rešenje:


 

\(x^2=t, x^4=t^2\)

 

\(t^2-\left ( a^2 +\frac{1}{a^2}\right )t+1=0\)

 

\(t_{1,2}=\frac{\frac{a^4+1}{a^2}\pm \sqrt{\left (\frac{a^4+1}{a^2}  \right )^2-4}}{2}\)

 

\(t_{1,2}=\frac{\frac{a^4+1}{a^2}\pm \sqrt{\frac{a^8+2a^4+1-4a^4}{a^4}}}{2}\)

 

\(t_{1,2}=\frac{\frac{a^4+1}{a^2}\pm \sqrt{\frac{(a^4-1)^2}{a^4}}}{2}\)

 

\(t_{1,2}=\frac{\frac{a^4+1}{a^2}\pm \frac{a^4-1}{a^2}}{2}\)

 

\(t_{1}=\frac{\frac{a^4+1}{a^2}+ \frac{a^4-1}{a^2}}{2}=\frac{2a^4}{2a^2}=a^2\)

 

\(t_{2}=\frac{\frac{a^4+1}{a^2}- \frac{a^4-1}{a^2}}{2}=\frac{2}{2a^2}=\frac{1}{a^2}\)

 

\(x^2=a^2\Rightarrow x_{1,2}=\pm a\Rightarrow x_{1}=a, x_{2}=-a\)

 

\(x^2=\frac{1}{a^2}\Rightarrow x_{3,4}=\pm \frac{1}{a}\Rightarrow x_{3}=\frac{1}{a}, x_{4}=-\frac{1}{a}\)