<s>[INST] 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 $a^2 + b^2$ ?

 [/INST] Using what you already did ( the discriminant is no-negative and thus $\;a^2-4b\ge8\;$) then
$$a^2+b^2\ge8+4b+b^2=(b+2)^2+4$$
So the above rightmost expression is at least...
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