combinatorics - Solving ${{2x - 3}choose{1- x^2}} = 3$

I was given this problem by my sister, she took it from a past paper in her calculus class (12 grade).

As the title reads, we are asked to solve the equation:

$${{2x - 3}\choose{1- x^2}} = 3$$

Now this seems a particularly strange problem to me. I see no way to solve this, provided that they haven't studied anything regarding binomial expansions with rational coefficients.

After going through the base conditions that $x$ should satisfy and trying some combinatorial identities I came empty handed. Wolfram Alpha provides a numerical solution that I don't see how to come by using pen and paper.

How would you solve this without making any use of any mathematical apparatus (i.e. gamma function) behind high school level calculus?