Trigonometric Substitution Maybe?

Algebra Level 5

$\large \frac { 1 }{ x } +\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } =\frac { 35 }{ 12 }$

What is the sum of all real roots of the equation above? Give your answer to 3 decimal places.

Details and Assumptions

You may only use a scientific calculator to determine the square root of a number.