2150.22 – Diminished by a Reciprocal

Let a real constant RR be given. Consider non-zero numbers NN such that when NN is diminished by 44 times its reciprocal, the result somehow equals RR. The value of any such NN, in terms of RR, is:

  1. 1/R1/R,
  2. RR,
  3. 44,
  4. 1/41/4,
  5. −R-R.
Solution

Since N–4N=R,N – \frac{4}{N} = R, we have

\begin{aligned}

N – \frac{4}{N} – R = 0, \

N^2 – 4 – R N = 0, \end{aligned}

which is a nice quadratic in NN, and

N=R±R2+162.N = \frac{R ± \sqrt{R^2 + 16}}{2}.

The sum of these two N’s is

R+R2+162+R−R2+162=R\frac{R + \sqrt{R^2 + 16}}{2} + \frac{R - \sqrt{R^2 + 16}}{2} = R

The answer is b.