$$Threetestscores=78,86,82$$

$$Threescoresweighted=20\mathrm{\%}=0.2$$

$$Finalscoresweighted=40\mathrm{\%}=0.4$$

$$Finalscores=x=?$$

So we have,

$$78(0.2)+86(0.2)+85(0.2)+0.4x\ge 86$$

$$246(0.2)+0.4x\ge 86$$

$$246(0.2)+0.4x\ge 86$$

$$49.2+0.4x\ge 86$$

$$x\ge \frac{36.8}{0.4}$$

$$\therefore x\ge 92$$