f(x) = 2x^2 - kx^2 + 4x + 5k - 2.
One root is the reciprocal of the other. Find the roots.
+130077 (2 - k)x ^2 + 4x + (5k - 2)
Call the roots m and 1/m
The product of the roots = (5k - 2) / (2 - k)
So
m * ( 1 /m) = (5k - 2) / (2 - k)
1 = (5k - 2) / (2 - k)
(2- k) = ( 5k - 2)
4 = 6k
k = 4 / 6 = 2 / 3
So
(2 - k)x^2 + 4x + (5k - 2) = 0
(2 - 2/3)x^2 + 4x + (5 *(2/3) - 2) = 0
(4/3)x^2 + 4x + 4/3 = 0
4x^2 + 12x + 4 = 0
4 ( x^2 + 3x + 1) = 0
x^2 + 3x + 1 = 0
x^2 + 3x = -1
x^2 + 3x + 9/4 = -1 + 9/4
(x + 3/2)^2 = 5/4
x + 3/2 = sqrt (5) / 2
x = [sqrt (5) - 3 ] / 2 = m
And
1/ m = 2 / [sqrt (5) - 3]
