Mathematics - Quadratic Equation Question with Solution | TestHub

For a quadratic expression $a{x}^2+bx+c$ to have the same sign as $a$ for all real $x$, its graph must either be entirely above the x-axis (if $a>0$) or entirely below the x-axis (if $a<0$). This means the quadratic equation $a{x}^2+bx+c=0$ must have no real roots. Therefore, its discriminant must be negative, i.e., $b^2-4ac<0$.