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 \(x^2 + bx + c = (x + b/2)^2 + c - b^2 / 4 > 0\)

Answer 1

stuck :-D

Answer 2

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

Answer 3

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

Answer 4

can't you just find vertex?

Answer 5

yea that works

Answer 6

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

Answer 7

*facepalm

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