Mathematics - Quadratic Equation Question with Solution | TestHub

For the quadratic expression $(p-1)x^2+(p-1)x+1$ to be positive for all real $x$, two cases are considered. Case 1: If $p-1=0$, i.e., $p=1$, the expression becomes $1$, which is always positive. Case 2: If $p-1\ne 0$, then the leading coefficient must be positive, $p-1>0⟹p>1$, and the discriminant must be negative, $D<0$. The discriminant is $(p-1{)}^2-4(p-1)(1)=(p-1)(p-1-4)=(p-1)(p-5)$. Setting $(p-1)(p-5)<0$ implies $1<p<5$. Combining both cases, $p=1$ and $1<p<5$, the solution is $1\le p<5$.