<s>[INST] The equation $y = x^2 + \frac{1}{2}ax + a$ represents a parabola for all real values of $a$ The task is to show that any parabola with the value a passes through shared point.
I graphed the parabola and understand that the shared point is $(-2,4)$. My current issue is generalizing why this is and how to prove it.
 [/INST] Just a note that $$x^2+\frac{1}{2}ax+(a-4)\\=x^2+2x-2x+\frac{1}{2}ax+a-4\\=x(x+2)-\left(2-\frac{1}{2}a\right)(x+2)\\=(x+2)\left(x-2+\frac{1}{2}a\right)$$
And hence, you get 2 points which equal  4 for the original function: $x=-2$ and $x=2-\frac{1}{2}a$.
In general, there is such common points when you have some expression $x^2+f(a) x+g(a)$.
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