help with equation

Let 

$a = x^2 - 10x - 45\\ b = x^2 + 10x - 29\\ c = x^2 - 10x - 69$

$\dfrac1a + \dfrac1b = \dfrac2c\\ 2ab = c(a + b)\\ 2(x^2 - 10x - 45)(x^2 + 10x - 29) = 2(x^2 - 10x - 69)(x^2 - 37)$

$(x^2 - 10x - 45)(x^2 + 10x - 29) = (x^2 - 10x - 69)(x^2 - 37)$

Then expand and simplify.

$5x^3 - 34x^2 - 265x - 624 = 0$

Solving by numerical methods gives $x\approx 12.055$.