3. (HUN) Let $x$ be an angle and let the real numbers $a, b, c, \cos x$ satisfy the following equation: 
$$ a \cos ^{2} x+b \cos x+c=0 . $$ 
Write the analogous quadratic equation for $a, b, c, \cos 2 x$. Compare the given and the obtained equality for $a=4, b=2, c=-1$. 
Second Day
3. Multiplying the equality by $4\left(a \cos ^{2} x-b \cos x+c\right)$, we obtain $4 a^{2} \cos ^{4} x+$ $2\left(4 a c-2 b^{2}\right) \cos ^{2} x+4 c^{2}=0$. Plugging in $2 \cos ^{2} x=1+\cos 2 x$ we obtain (after quite a bit of manipulation):  
$$ a^{2} \cos ^{2} 2 x+\left(2 a^{2}+4 a c-2 b^{2}\right) \cos 2 x+\left(a^{2}+4 a c-2 b^{2}+4 c^{2}\right)=0 $$  
For $a=4, b=2$, and $c=-1$ we get $4 \cos ^{2} x+2 \cos x-1=0$ and $16 \cos ^{2} 2 x+8 \cos 2 x-4=0 \Rightarrow 4 \cos ^{2} 2 x+2 \cos 2 x-1=0$.