Question

The integral values of $m$ for which the roots of the equation $m{x}^2+(2m-1)x+(m-2)=0$ are rational are given by the expression [where $m$ is integer]

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

Answer 1

The correct option is C $n(n+1)$

Given that:

$m{x}^2+(2m-1)x+(m-2)=0$ where $m$ is an integer.

To find:

$m=?$

Solution:

$Discriminant,D=b^2-4ac$

or, $D=(2m-1{)}^2-4m(m-2)$

or, $D=4m^2+1-4m-4m^2+8m$

or, $D=4m+1$

For a rational root, $D$ must be a square.

Since value of $D$ is odd number,

So $D=4m+1$ can be represented as

Let $2n+1$ is an odd number

or, $(2n+1{)}^2=4n^2+4n+1$

$=4n(n+1)+1$

So integral value of $n$ is $n(n+1).$

Hence, C is the correct answer.