9.1. It is known that $a+b+c<0$ and that the equation $a x^{2}+b x+c=0$ has no real roots. Determine the sign of the coefficient $c$.

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Answer: $c<0$.

Solution. The quadratic trinomial $f(x)=a x^{2}+b x+c$ has no real roots, which means it maintains the same sign for all values of the argument x. Since $f(1)=a+b+c<0$, then $f(0)=c<0$.