Asked by Shawn
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.
Answers
Answered by
Steve
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.
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.
Answered by
Steve
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.
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.
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