Minimum value of $a^2 + b^2$ so that the quadratic $x^2 + ax + (b+2) = 0$ has real roots 
The equation $ x^2 + ax + (b+2) = 0 $ has real roots, where $a$ and $b$ are real numbers. 
How would I find the minimum value of $a^2 + b^2$ ?


In the $(a,b)$ plane, the point $(a,b)$ has to be under the parabola $a^2-4b =8$. The point of this domain closest to origin  is the vertex of the parabola, $(0,-2)$. Hence the minimum of $a^2+b^2$ is $4$.
