Profit = revenue - cost.
x items are sold for xP=200x-0.01x^2.
The cost of producing x items is C=50x+20000.
Profit=200x-0.01x^2-50x-20000=150x-0.01x^2-20000.
To maximise the profit we differentiate this: 150-0.02x and equate to zero, so x=150/0.02=50*150=7500.
To solve without using calculus:
150x-0.01x^2-20000=-(0.01x^2-150x+20000)=
-0.01(x^2-15000+2000000)=
-0.01(x^2-15000+7500^2+2000000-7500^2)=
-0.01((x-7500)^2+2000000-56250000)=-0.01((x-7500)^2-54250000)=
0.01((54250000-(x-7500)^2).
This expression has the greatest value when x=7500.