Zadatak 0230

Test: MKsIIJAII_047

 

Reši po x jednačinu (a i n su realni parametri):

 \(\frac{a}{nx-x}-\frac{a-1}{x^2-2nx^2+n^2x^2}=1\).

 

Rešenje:


 

\(\frac{a}{x(n-1)}-\frac{a-1}{x^2(n^2-2n+1)}=1\)

\(\frac{a}{x(n-1)}-\frac{a-1}{x^2(n-1)^2}=1\)

\(\frac{ax(n-1)-(a-1)}{x^2(n-1)^2}=1\)

\(x^2(n-1)^2=ax(n-1)-(a-1)\)

\(x^2(n-1)^2-ax(n-1)+(a-1)=0\)

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

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

\(x_{1,2}=\frac{a(n-1)\pm (n-1)(a-2)}{2(n-1)^2}\)

\(x_{1,2}=\frac{a\pm (a-2)}{2(n-1)}\)

\(x_{1}=\frac{2a-2}{2(n-1)}\)   i   \(x_{2}=\frac{a- a+2}{2(n-1)}\)

\(x_{1}=\frac{a-1}{n-1}\)     i     \(x_{2}=\frac{1}{n-1}\).