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 $a^2 + b^2$ ?


We get the solutions
$$
0 = x^2 + a x + b + 2 = (x + a/2)^2 - a^2/4 + b + 2 \iff \\
x = \frac{-a \pm \sqrt{a^2 - 4b - 8}}{2}
$$
which are real for
$$
a^2 - 4b - 8 \ge 0
$$
We now want to solve the optimization problem
\begin{matrix}
\min & a^2 + b^2 \\
\text{w.r.t} & a^2 - 4b - 8 \ge 0
\end{matrix}

The above image shows the function
$$
z = f(x,y) = x^2 + y^2
$$
(red surface). It is rotational symmetric around the $z$-axis and has circles in the $x$-$y$-plane with center $(0,0)$ and radius $\sqrt{c}$ as isolines $f(x,y) = c$ (yellow). 
The curve 
$$
x^2 - 4y - 8 = 0 \iff \\
y = (1/4) x^2 - 2 
$$
(green) is the border of the feasible area $x^2 - 4y -8 \ge 0$, shown in light blue. Point $A=(2,0)$ (pink) is not feasible with the condition, $B=(2,-2)$ (blue) is a feasible point.
Going from inner to outer isolines, we see that the isoline with radius $r=2$, thus $x^2 + y^2 = 2^2 = 4 = c$ is the first one to be part of the feasible area. Thus the minimum is $4$.
