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. 1R\dfrac{1}{R}

  2. RR

  3. 44

  4. 14\dfrac{1}{4}

  5. −R-R

Solution

Since N−4N=RN - \dfrac{4}{N} = R, we have

N−4N−R=0N2−4−RN=0\begin{aligned}N - \dfrac{4}{N} - R &= 0 \\N^2 - 4 - R N &= 0\end{aligned}

which is a nice quadratic in NN, and

N=R±R2+162N = \dfrac{R \pm \sqrt{R^2 + 16}}{2}

The sum of these two N’s is

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

The answer is b.