<s>[INST] Find the range of possible values of $\sqrt{a^2+a+1}-\sqrt{a^2-a+1}$ 
Let $x=\sqrt{a^2+a+1}-\sqrt{a^2-a+1},x\in \mathbb R$. Find range of possible values of $x$.

I tried drawing the graph and obtained this:
Through which the answer came out to be $(-1,1)$.
What should be the procedure through algebra?
 [/INST] Square both sides to get
$$
x^2 = 2(a^2 + 1  - \sqrt{a^4 + a^2 + 1})
$$
We can then use $a^4 + a^2 + 1 > (a^2 + 1/2)^2$ to see that $x^2 < 1$. Now rearrange and square again to get
$$
(x^2 - 2a^2 - 2)^2 = 4(a^4 + a^2 + 1).
$$
Assuming $|x| < 1$, we can solve for $a$ to get
$$
a =x\sqrt{\frac{1- (x/2)^2}{1-x^2}}.
$$
This is defined for every $x \in (-1,1)$, so $x(a)$ covers the entire range $(-1,1)$.
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