Find the smallest possible value of $ x^2 + bx + c $ and of $ ax^2 + bx + c $ for a 0

I've shown by completing the square that $x^2 + bx + c = (x + b/2)^2 + c - b^2 / 4 0$

Question

Find the smallest possible value of $ x^2 + bx + c $ and of $ ax^2 + bx + c $ for a > 0

I've shown by completing the square that $x^2 + bx + c =  (x + b/2)^2 + c - b^2 / 4 > 0$

anonymous · Answer

stuck :-D

anonymous · Answer

for $ax^2 + bx + c = a(x^2 + b/a x + c/a) = a((x + b/2a)^2 - b^2/4a + c > 0)$

anonymous · Answer

(x+b/2)2 is either 0 or positive.

anonymous · Answer

can't you just find vertex?

anonymous · Answer

yea that works

anonymous · Answer

what's the vertex denoted by the above two equations (after I completed their square)

anonymous · Answer

*facepalm

seems to be c−b2/4 for the first one, and −b2/4a+c for the second one