Mathematics - Quadratic Equation Question with Solution | TestHub

For the roots of a quadratic equation $A{x}^2+Bx+C=0$ to lie on opposite sides of the origin (i.e., $x=0$), the condition is $A\cdot f(0)<0$. Here, $A=m^2-1$ and $f(0)=1$. So, we must have $(m^2-1)(1)<0$, which implies $m^2-1<0$. Factoring, $(m-1)(m+1)<0$, which gives $m\in (-1,1)$. We must also check the cases where $m^2-1=0$. If $m=1$, the equation becomes $1=0$, which has no roots. If $m=-1$, the equation becomes $-4x+1=0$, which has a single root $x=1/4$, not roots on opposite sides of the origin. Thus, the valid range for $m$ is $(-1,1)$.