Mathematics - Quadratic Equation Question with Solution | TestHub

The discriminant of $f(x)$ is $D=(-(m+2){)}^2-4(1)(2m)=m^2+4m+4-8m=m^2-4m+4=(m-2{)}^2$. For roots to be real and distinct, $D>0$, so $(m-2{)}^2>0$, which means $m\ne 2$. Thus, A is incorrect. For both roots to be positive, we need $D\ge 0$ (which is always true as $(m-2{)}^2\ge 0$), sum of roots $m+2>0⟹m>-2$, and product of roots $2m>0⟹m>0$. Combining these, we need $m>0$. If $m\in [4,\infty )$, then $m>0$, so both roots are positive. Thus, B is correct. For one root to be positive and the other negative, the product of roots must be negative: $2m<0⟹m<0$. So, C is correct. For both roots to be greater than 1, we need $D\ge 0$ (always true), $f(1)>0$, and the axis of symmetry $x=\frac{m+2}{2}>1$. $f(1)=1-(m+2)+2m=m-1>0⟹m>1$. $\frac{m+2}{2}>1⟹m+2>2⟹m>0$. Combining these, we need $m>1$. If $m\in (4,\infty )$, then $m>1$, so both roots are greater than 1. Thus, D is correct.