Mathematics - Quadratic Equation Question with Solution | TestHub

For a quadratic expression $f(x)=a{x}^2+bx+c$ to have the same sign as $a$ for all real $x$, the parabola must not intersect or touch the x-axis. This means the quadratic equation $a{x}^2+bx+c=0$ has no real roots. Therefore, its discriminant $D=b^2-4ac$ must be negative, i.e., $b^2-4ac<0$.