Zadatak 0237

Test: MPsIIJAII_049

 

Reši jednačinu:

 

\(b^2x^4-(b^4+a^2)x^2+a^2b^2=0\)

 

Rešenje:


 

\(b^2x^4-(b^4+a^2)x^2+a^2b^2=0\)

 

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

 

\(b^2t^2-(b^4+a^2)t+a^2b^2=0\)

 

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

 

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

 

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

 

\(t_{1}=b^2, t_{2}=\frac{a^2}{b^2}\)

 

\(x^2=a^2, x^2=\frac{a^2}{b^2}\)

 

\(x_{1,2}=\pm a, x_{3,4}=\pm \frac{a}{b}\)

 

\(x_{1}=a, x_{2}=-a, x_{3}= \frac{a}{b}, x_{4}=-\frac{a}{b}\).