Quadratic Function: possible value of c

Question

I have $f(x)=x^2-2bx+c$ If the minimum value of the function is 6, what is the possible value of c.
I tried$$f(x)=(x-b)^2-b^2+c$$ $$b^2=c-6$$ I couldn't solve for the value of c.

Answer 1

The vertex is where the min/max occurs. In this case, it's the min. Thus $x = -\dfrac{-2b}{2} = b \implies f_{\text{min}} = f(b) = b^2-2b\cdot b+c = c-b^2 = 6\implies c = b^2+6$.