6. Zadatak

Pojednostavnimo lijevu stranu jednadžbe:

$$\begin{aligned} \frac{x^4-4x^3+8x^2-16x+16}{x^4-16} &=\frac{(x^2+4)^2-4x^3-16x}{(x-2)(x+2)(x^2+4)}\\ &=\frac{(x^2+4)^2-4x(x^2+4)}{(x-2)(x+2)(x^2+4)}\\ &=\frac{(x^2+4)(x^2+4-4x)}{(x-2)(x+2)(x^2+4)}\\ &=\frac{(x-2)^2}{(x-2)(x+2)}=\frac{x-2}{x+2}. \end{aligned}$$

Pojednostavnimo desnu stranu jednadžbe:

$$\begin{aligned} \frac1{1\cdot2}+\frac1{2\cdot3}+\frac1{3\cdot4}+\cdots+\frac1{25\cdot26} &=\frac11-\frac12+\frac12-\frac13+\frac13-\frac14+\cdots+\frac1{25}-\frac1{26}\\ &=1-\frac1{26}=\frac{25}{26}. \end{aligned}$$

Zadatak svodimo na rješavanje jednadžbe

$$\frac{x-2}{x+2}=\frac{25}{26} \quad\Longrightarrow\quad 26x-52=25x+50 \quad\Longrightarrow\quad x=102.$$