Mathematics - Quadratic Equation Question with Solution | TestHub

The condition that the quadratic equation $p{x}^2+qx+r=0$ has no real roots means its discriminant $D=q^2-4pr<0$. When the discriminant is negative, the quadratic expression $f(x)=p{x}^2+qx+r$ has the same sign as the leading coefficient $p$ for all real values of $x$. The expression $p+q+r$ is simply the value of $f(x)$ when $x=1$, i.e., $f(1)$. Since $f(x)$ always has the same sign as $p$, it follows that $f(1)=p+q+r$ must also have the same sign as $p$.