Shifted Exponential Calibration from Raw Moments

A toy latency model assumes $$X = c + Y, \qquad Y \sim \mathrm{Exp}(\lambda),$$ with unknown deterministic floor $c>0$ and unknown rate $\lambda$.

From historical data, the empirical mean of $X$ is $8$ and the empirical second raw moment is $73$.

Use the method of moments to estimate $c$ and $\lambda$.