1.
Let R(p)=Ap²+Bp+C and E(p)=Dp+E where A, B, C, D, E are constants.
At breakeven, R(p)=E(p) so Ap²+Bp+C=Dp+E.
But we know F(p-43)(p-97)=0=Fp²-140pF+4171F.
And we know 51823=A(43)²+43B+C=43D+E and 17499=A(97)²+97B+C=97D+E.
So (51823-17499)=-54D and D=-34324/54=-17162/27=-635.63 approx.
But C=0 because R(p)=pq where q(p) is the quantity dependent on price making pq a polynomial without a constant term.
A(43)²+43B=51823 and A(97)²+97B=17499.
So 19353A+4171B=5026831, 404587A+4171B=752457.
(404587-19353)A=(752457-5026831), 385234A=-4274374, A=-11.10 approx.
Therefore B=1682.29 approx.
R(p)=p(-11.1p+1682.29), E(p)=-635.63p+79155.07.
2.
E=R at breakeven, so -20000p+90000=-2170p²+87000p.
2170p²-107000p+90000=0, p=48.45 and 0.86 approx.
E(0.86)=R(0.86)=72880.38. When p=48.45, E, R are negative.
x intercepts for R are 0 and 87000/2170=40.09 approx.
3.
Max R at p=19000/1480=12.84 approx., max R=125,255.66 approx.
Breakeven when 720p²-25500p+300000=0. No real solutions to this.