Question 1130474
P= -0.2N^2 + 3.6N - 9 
This is a quadratic with a=-0.2, b=3.6 and c= -9
The n value at the vertex is -b/2a or -3.6/-0.4, which is +9
so when n=9, the profit is maximum
put that back into the equation
P=-0.2*81+32.4-9, or 7.2 thousands or $7200.
setting the equation equal to zero will give the two break even points.
can use the quadratic equation
n=(-1/.4)(-3.6+/- sqrt (12.96-4(-0.2)(-9));that is sqrt of (12.96-7.2) or sqrt (5.76), which is 2.4
n=-2.5(-3.6+2.4) which is 3
n--2.5(-3.6-2.4) which is 15
That makes sense, because both are six units from the vertex/maximum, consistent with symmetry that quadratics have around the vertex.

Can also multiply everything by -5 to make the equation n^2-18n+45=0
That factors into (n-15)(n-3)=0, and n=3, n=15

{{{graph(300,300,-10,20,-10,10,-.2x^2+3.6x-9)}}}