@kevin4849 In the context of Quadratic functions, irrational and imaginary roots do come in pairs, both the + and - roots. The same would go with 4th roots, there were be two sets of pairs. This doesn't apply to x^2 = 0......Also, depending on context, roots may be extraneous.
Irrational roots don't always come in pairs. For example, the polynomial x^3-2=0 has only one root which is the cube root of 2. Here's my question: do even-numbered irrational roots come in pairs? In other words, roots of the form A+B*(nth root of) C always come in pairs, when n is positive and even?
If you're doing the AP calc exam, an optimization like this where you have to find such a lengthy derivative prob won't be on the MCQ section. Unless College Board decides to cuck all of us ofc...
it's always good to work through a problem like this in the Detailed Way. It's paying our Dues as we learn how these problems are solved. Eventually, we discover other ways to solve this problem. That second way is realizing we have Similar Triangles. That leads to a General Formula where x = (h1* R) / (h1 + h2) .. R = the distance Between the Poles, 48 in this problem. Check it out, it works.. Thank you for this classic Calculus Min/Max problem :)
