Let's expand the left side of the equation:

(x-10)^2 = x^2 - 20x + 100
(x+9)^2 = x^2 + 18x + 81

Now, we can rewrite the equation with the expanded terms:

(x^2 - 20x + 100) + (x^2 + 18x + 81) = 2x^2

Now, combine like terms:

2x^2 - 2x + 181 = 2x^2

Subtract 2x^2 from both sides:

-2x + 181 = 0

Add 2x to both sides:

181 = 2x

Divide by 2:

x = 90.5

Therefore, the solution to the equation is x = 90.5.