Question

Form a quadratic equation whose roots are k and 1/k​

Answer 1

Answer:

Generally, without going into semantics, if a function f(x) has root r , then f(x)=(x−r)g(x) , where g(x) has a degree which is one smaller than the degree of f(x) .

Applying this result twice with r=1 and r=1/2 gives f(x)=(x−1)(x−1/2)g(x) .

However, the clue is in the question - that is that we want f(x) to be a quadratic - I.e. the degree of f(x) should be 2, I.e. the largest power of an x should be 2. Since the x ’s in f(x) already give an x2 term, g(x) cannot have an x in it, otherwise there would be an x3 value in f(x) , and therefore f(x) would not be quadratic. It follows that g(x) must be a (nonzero) constant, just a number - so we can say g(x)=c .

For the sake of easy numbers, say g(x)=1 , then f(x)=(x−1)(x−1/2) is a quadratic equation with roots 1 and 1/2.

This answer may seem too detailed, however this is a pure mathematically minded approach, and can easily be generalised to answer any “what is the quadratic/cubic/quartic/dodecahectic equation with roots x,y,z, etc.?”