A baker determined the annual profit in dollars from selling pies using p n( ) = 52n − 0.05n2 , where n is the number of pies so
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Sounds like you're asked to find 
such that
In other words, find that satisfies
We can factorize this as
In order that
describes a probability distribution, require that 
for all 
, which means we can ignore the possibility of 
.
Let
.
Multiply both sides by
.
We want to find
that removes the quartic and quadratic terms from the equation, i.e.
so the cubic above transforms to
Substitute
and we get
![\implies y=\sqrt[3]{\dfrac{9+\sqrt{93}}{18}}](https://tex.z-dn.net/?f=%5Cimplies%20y%3D%5Csqrt%5B3%5D%7B%5Cdfrac%7B9%2B%5Csqrt%7B93%7D%7D%7B18%7D%7D)
![\implies a=\sqrt[3]{\dfrac{9+\sqrt{93}}{18}}-\dfrac13\sqrt[3]{\dfrac{18}{9+\sqrt{93}}}](https://tex.z-dn.net/?f=%5Cimplies%20a%3D%5Csqrt%5B3%5D%7B%5Cdfrac%7B9%2B%5Csqrt%7B93%7D%7D%7B18%7D%7D-%5Cdfrac13%5Csqrt%5B3%5D%7B%5Cdfrac%7B18%7D%7B9%2B%5Csqrt%7B93%7D%7D%7D)
In other words, find
We can factorize this as
In order that
Let
Multiply both sides by
We want to find
so the cubic above transforms to
Substitute