The integral value of for which the root of the equation `m x^2+(2m-1)x+(m-2)=0` are rational are given by the expression [where `n` is integer]

Options

A. `n^(2)`
B. `n(n + 2)`
C. `n(n + 1)`
D. none of these

Answer

3

Explanation

Discriminant `D = (2m - 1)^(2) - 4(m - 2)m = 4m + 1` must be perfect square. Hence,
`4m + 1 = k^(2)`, for some `k in I`
`rArr m = ((k - 1)(k + 1))/(4)`
Clearly, k must be odd. Let k = 2n + 1.
`therefore m = (2n(2n + 2))/(4) = n(n + 1), n in I`