There might be some shortcut I'm not aware of but the brute force method of solving this isn't trivial and is a great exercise.
The 30 second summary is that by taking the sum of your two random variables you create a third one
\(\displaystyle \hat{X} = X_1+X_2\) that has a new distribution.
You then need to find the interval about the mean of this new random variable that integrates to 90%.
So the first step is to find the distribution of \(\displaystyle \hat{X}\)
You might read up on the distribution of a sum of independent rv's being the convolution of their individual distributions.
Each of your \(\displaystyle X_i\)s is identically distributed and quadratic in x. Their convolution won't be too bad.
Find the mean of \(\displaystyle \hat{X}\) and then find the interval about it that integrates to 90%.
One might be concerned that there are different intervals that satisfy this if the distribution isn't symmetric. In this case the distribution will be symmetric about it's mean because the underlying distribution is and addition is a linear function.
