4.2 Quadratic equations and inequations — MATHS STD 11 Science — Question

If $c > 0$, then by hypothesis $f(x) > 0$ This means that the curve $y = f(x)$ does not meet $x$-axis.

If $c < 0$, then by hypothesis$f(x) < 0$, which means that the curve $y = f(x)$ is always below $x$-axis and so it does not intersect with $x$-axis.

Thus in both cases $y = f(x)$ does not intersect with $x$-axis i.e. $f(x) \ne 0$for any real $x$.

Hence $f(x) = 0$i.e. $a{x^2} + bx + c = 0$ has imaginary roots and so${b^2} < 4ac$.