The sum of a number and its reciprocal is equal to 4 times the reciprocal, What is the number?

3 Answers
Sep 11, 2017

There are two possibilities:

x=-sqrt(3)x=3 or x=sqrt(3)x=3

Explanation:

Suppose the number is xx

Then the "reciprocal" of this number is 1/x1x

We are told that:

"The sum of a number and its reciprocal is equal to 4 times the reciprocal"

Therefore:

x + 1/x = 4 xx 1/x x+1x=4×1x

:. x = 4 xx 1/x - 1 xx 1/x

:. x = 3 xx 1/x

:. x^2 = 3

:. x = +-sqrt(3 )

Sep 11, 2017

n + 1/n = 4 * 1/n

on the left side, multiply n by n/n (same as multiplying by 1) to get a common denominator so you can add the numerators...

n^2 /n + 1/n = 4/n

giving:

n^2 + 1 = 4

n^2 = 3

so...

n = +-sqrt(3)

...check your work, when you can:

sqrt(3) + 1/sqrt(3) = 4 * 1/sqrt(3)

sqrt(3)*sqrt(3)/sqrt(3) + 1/sqrt(3) = 4/sqrt(3)

3/sqrt(3) + 1/sqrt(3) = 4/sqrt(3)

4/sqrt(3) = 4/sqrt(3)

(the same sequence also applies for the -sqrt(3) case

Sep 11, 2017

I got two possible numbers: +-sqrt(3)

Explanation:

Let us call the unknown number x; we get:
x+1/x=4*1/x
rearrange:
x^2+1=4
(with x!=0)
x^2=3
x=+-sqrt(3)