Reciprocal Equation

If a non-zero number N, diminished by 4 times its reciprocal, equals a given real constant R, then in terms of R, the sum of all such possible values of N is:

  1. 1/R
  2. R
  3. 4
  4. 1/4
  5. R


Show/Hide Solution

Since N –  4  = RN –  4  – R = 0 → N2 − 4 − RN = 0 → N2RN − 4 = 0.
N N

So we have a nice quadratic in N, and

N R ± √R2 + 16 .
2

So

N R + √   or  N R− √  .
2 2

The sum of the N‘s is

R + √   +  R − √   =  R  +     +  R  −     =  2R  = R.
2 2 2 2 2 2 2

The answer is (b).

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