How do I solve this without using a scientific calculator: $3-x^2 = 2^x$

Question

Is there any way to solve the equation $3-x^2 = 2^x$ without using the calculator to estimate the solution for you? So far I've tried changing the base of the right hand side to base $e$ so $3-x^2 = e^{x\ln2}$ ,but the equation still cant be isolated for x?
$$\frac{\ln(3-x^2)}{x} = \ln2$$
Am I missing something crucial here?

Answer 1

HINT: $\ln 3^{-x^2}=-x^2\ln 3$, and $\ln 2^x=x\ln 2$. Let $a=\ln 3$ and $b=\ln 2$; then after taking the natural log on both sides you have $-ax^2=bx$. Can you solve that for $x$?

Once you've found its solutions in terms of the constants $a$ and $b$, you can always substitute the correct numbers for $a$ and $b$.