Pismeni zadaci

Zadatak 0236

Test: MPsIIJAII_049

 

Reši jednačinu:

 

\(\frac{2x-5}{x-11}+\frac{7-3x}{(x-6)^2+1}=\frac{x-4}{x-11}+\frac{(x-3)^2}{(x-6)^2+1}\)

 

Rešenje:


 

\(\frac{2x-5}{x-11}-\frac{x-4}{x-11}-\frac{(x-3)^2}{(x-6)^2+1}+\frac{7-3x}{(x-6)^2+1}=0\)

 

\(\frac{x-1}{x-11}+\frac{-x^2+6x-9+7-3x}{(x-6)^2+1}=0\)

 

\(\frac{(x-1)(x^2-12x+37)+(x-11)(3x-x^2-2)}{(x-11)((x-6)^2+1)}=0\)

 

\(\frac{x^3-12x^2+37x+12x-37 +3x^2-33x-x^3+11x^2-2x+22}{(x-11)((x-6)^2+1)}=0\)

 

\(\frac{x^2+14x-15}{(x-11)((x-6)^2+1)}=0\)

 

\(\Leftrightarrow x^2+14x-15=0\wedge (x-11)((x-6)^2+1)\neq 0\)

 

\(x_{1,2}=\frac{-14\pm \sqrt{196+60}}{2}\wedge x\neq 11,(x-6)^2\neq -1\)

 

\(x_{1,2}=\frac{-14\pm 16}{2}\)

 

\(x_{1}=1, x_{2}=-15\).

 


 

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