Values of $a$ for which the equation $100^{-\lvert x \rvert} - x^2 = a^2$ has the maximum amount of solutions

Question

$100^{-\lvert x \rvert} - x^2 = a^2$

I don't know how to approach this problem, due to the x in the exponent. I would appreciate hints more than outright solutions :)

Answer 1

I am assuming you are looking for solutions in real numbers. Think in terms of the intersection of two curves. $y=x^2+a^2$ (a parabola which lies on or above the $x-$axis) and $y=100^{-|x|}$ (even function). Now controlling $a$ gives you the ability to move your parabola up and down. Try to see what range of $a$ can help you maximize the number of intersections of the two curves.

See the picture: