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Q.
Given that $ax^2 + bx + c = 0$ has no real roots and $a + b + c < 0$, then
Complex Numbers and Quadratic Equations
Solution:
Since the equation $ax^2 + bx + c$ = 0 has no real roots.
$ \therefore $ the curve $y = ax^2 + bx + c$ does not meet $x$-axis.
$\therefore \, f(x) = ax^2 + bx + c$ has the same sign for all values of $x$.
But $a + b + c < 0$ and $f(1) = a + b + c < 0$
$ \therefore \, f(x) < 0$ for all $x$,
$\therefore \, f(0) < 0 $
$\Rightarrow c < 0 $