<s>[INST] Solving $\sqrt{9-x^2} > x^2 + 1$ without graphic calculator for the exact form 
Is there any way to solve this inequality without using a graphic calculator to get the exact form?
$$\sqrt{9-x^2} > x^2 + 1$$

I've tried completing the square but I end up with
$$\frac{3 - \sqrt{41}}{2} < x^2 < \frac{3 + \sqrt{41}}{2}$$
which does not match with the answer on Desmos.
 [/INST] $$9-x^2>x^4+2x^2+1$$
$$x^4+3x^2-8<0$$
$$\left(x^2+\frac32 \right)^2 < 8+\frac94$$
$$\left(x^2+\frac32 \right)^2< \frac{41}4$$
$$0\le x^2<\color{red}-\frac32 + \frac{\sqrt{41}}2$$
Now, your answer should coincide.
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