Question 185
Determine all triplets of real numbers $(a, b, c)$, with $a \neq b$, $a \neq 0$, $b \neq 0$, such that the parabolas $$
y = a x^{2} + b x + c, \quad y = b x^{2} + c x + a
$$ have the same vertex.
y = a x^{2} + b x + c, \quad y = b x^{2} + c x + a
$$ have the same vertex.