Determine the nonzero real number $c$ having the property that $f(c) = a$ is a relative minimum of $f(x) = x^2 - ax - c$.
My trouble with this problem is that I substituted c into the function, and by simplifying, I got
c^2-a(c+1)-c=0
I don't know where I should go from here.
The minimum is attained at $x=c \;\; $ if $$f'(c)=2c-a=0$$ thus $$c=\frac {a}{2} .$$ The condition
$$f (c)=a$$ gives $$\frac {a^2}{4}-a\frac {a}{2}-\frac {a}{2}=a $$
and $$ a (\frac {3}{2}+\frac {a}{4})=0.$$
Finally $$c\in \{-3,0\} $$