how can I solve the following equation (without complex numbers)? $$
\sqrt{45-x^2} = 3- x^2
$$
$$
\sqrt{13-x^2} = 7 - x^2
$$
I have tried the Quadratic formula.  I always came up with the solution of $\{-3, 3\}$ by the first one and $\{-3, 3\}$ by the second one which is both wrong. Does this happen because I have to pay attention to a rule when substituting in a root, which I don't know, or how should I arrive at the solution that the first has none according to the graph and the second has the solution set according to the graph $\{-2,2\}$? I search a lot in the internet but there was no solution.

Consider the first equation:
A general strategy would be to square both sides:
$$\sqrt{45-x^2} = 3- x^2 \implies 45-x^2 = (3-x^2)^2 = 9+x^4-6x^2 \iff x^4-5x^2-36=0$$
Using $a=x^2$, the equation becomes $a^2-5a-36=0$ with $a\ge 0$. The Quadratic Formula indeed say that the solutions are $a=9$ and $a=-4$. Since $a\ge 0$, only $a=9$ is solution, so $x=\pm 3$.
But! The first step of computation ($\sqrt{45-x^2}=3-x^2 \implies 45-x^2=(3-x^2)^2$) is an implication, not an equivalence, so it is important to check that $3$ and $-3$ are indeed solution. As you check, they are not, so the equation has no real solution.
What are $3$ and $-3$ doing here? The equation $45-x^2=(3-x^2)^2$ is actually equivalent to $3-x^2=\pm\sqrt{45-x^2}$ (assuming $45-x^2\ge 0$). $4$ and $-3$ are solutions of the "twin" equation $3-x^2=-\sqrt{45-x^2}$.

Other method to see this equation has no solution: $3-x^2=\sqrt{45-x^2}$ implies that $3-x^2\ge 0$ so $x^2\le 3$, hence $3-x^2\le 3$. This in turn implies that $45-x^2\ge 42$ so $\sqrt{45-x^2}\ge\sqrt{42}>3\ge 3-x^2$ so the equation has no real solution.
