Question

Given that ax^2+bx+c=0 has no real roots and a+b+c<0. Then,
(a) c=0 (b) c>0 (c) c<0 (d) c not equal to 0

Answer 1

What are the consequences of that function having no real roots? Think about the quadratic equation. What must be true for the answer to not be real?

Answer 2

For having no real roots b^2 should be less than 4ac

Answer 3

$$a+b+c=a(1)^2+b(1)+c=f(1)$$ so we know \(f(1)<0\) and the fact that \(f\) has no real roots suggests that it never changes sign (by the intermediate value theorem). it is thus clear that \(f<0\) always and thus \(f(0)=a(0)^2+b(0)+c=c<0\). Q.E.D.

Answer 4

Thanks...