Question

Two positive numbers x and 1-x add up to 1. Let the sum of their reciprocals be k. Find the only possible value of k such that x is a rational number. Hint: set up the equation and solve x with k in it. You'll encounter the quadratic formula in your solution.

Answer 1

so, did you follow the hint?

1/x + 1/(1-x) = k
kx(1-x) - 1 = 0
kx^2 - kx + 1 = 0

Now, you know the roots are

(k±√(k^2-4k))/2

For the roots to be rational, you must have

k^2-4k

a perfect square.

See what you can do with that.

Answer 2

Actually, the problem is misstated. You need k to be an integer. Otherwise, pick any positive rational number for x, and you have a different value for k.

1/3 + 2/3 = 1
3 + 3/2 = 11/2 = k

3/17 + 14/17 = 1
17/3 + 17/14 = k

But those k values are not integers.