Find the positive solution to

$\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0$
We could clear out the denominators by multiplying, though that would be unnecessarily tedious.
To simplify the equation, substitute $a = x^2 - 10x - 29$ (the denominator of the first fraction). We can rewrite the equation as $\frac{1}{a} + \frac{1}{a - 16} - \frac{2}{a - 40} = 0$. Multiplying out the denominators now, we get:
\[(a - 16)(a - 40) + a(a - 40) - 2(a)(a - 16) = 0\]
Simplifying, $-64a + 40 \times 16 = 0$, so $a = 10$. Re-substituting, $10 = x^2 - 10x - 29 \Longleftrightarrow 0 = (x - 13)(x + 3)$. The positive [root](https://artofproblemsolving.com/wiki/index.php/Root) is $\boxed{013}$.