Question

Show that the equation x^2 = e^x + 1 has at least one solution

Answer 1

I find the wording here very interesting. It doesn't seem to be requiring you to FIND the solution, only to show that one exists.

If you look at a graph of the two functions, you can see that a solution DOES exist:
https://www.desmos.com/calculator/tg4e5mmsav

So without actually finding it, I think you can show that it exists by showing, for example, that there is a value x=a such that:

a^2 > e^a + 1 (like maybe a=-2)

And there is a value x=b such that:

b^2 < e^b + 1 (like maybe b=0)

Then, since the functions are both continuous, they have to have a point of intersection between a and b, and hence, that point is a solution to the equation.

Answer 2

Thank you so much! I really appreciate it :)