Solve $h(x)+h'(x)(8-x)-32=0$ for $x$.Where
$$h(x)=\frac{\frac{1}{16}x^2 - 2 x + 80}{\left(\frac{1}{16}x^2 - 2 x + 20\right)^2}$$
Should I go with characteristic equations? or is there another way.
The context for this question is somewhat involved - there is an ant walking up a hill (given by h(x)), at what point will he see the blade of grass given by the points (32,1/5)(32,8). His line of sight, given by h'(x) cannot interest h(x) for the hill is an obstruction. The first point the ant must see on of the grass must be point (32,8), for after h'(x) has reached a maximum as the ant walks up the hill it will decrease to zero, at the top of the hill. Therefore it will approach (32,8) from above. $$32=h'(x)*8+b$$ $$h(x)=h'(x)*x+b$$
Letting $g(x)=x^2/16-2x+20$, simplifies your approach a lot.