<s>[INST] If a and b are two distinct real values such that $F (x)= x^2+ax+b $ And given that $F(a)=F(b)$ ; find the value of $F(2)$ 
If $a$ and$ b$ are two distinct real values such that 
  $$F (x)= x^2+ax+b $$
   And given that  $F(a)=F(b)$ ; find the value of $F(2)$


My try:
$$F(a)=a^2+a^2+b= 2a^2+b,\quad
F(b)=b^2+ab+b $$
$F(a)=F(b)$  implies   $2a^2=b^2+ab$, and thus
$F(2)= 4+2a+b $
What Now?  
Any help would be appreciated , thank you 
 [/INST] Due to the symmetry of parabolas of that form, you can see $\frac{|a+b|}{2}$ is a minimum or maximum.
This gives you $f'\left(\frac{|a+b|}{2}\right)=|a+b|+a=0$ which implies $b=0\lor 2a+b=0$.
$b=0 \implies F(b)=0 \land F(a)=2a^2$
But $F(a)=F(b)\implies a=0$ which leads to a contradiction (as $a$ and $b$ are distinct).
Hence $2a+b=0$ as required.
</s>