Question
Let $$a > 0, b > 0$$ and $$c > 0$$ . Then the roots of the equation $$a{x^2} + bx + c = 0$$
A.
are real and negative
B.
have negative real parts
C.
both (A) and (B)
D.
none of these
Answer :
both (A) and (B)
Solution :
As $$a, b, c > 0, a, b, c$$ should be real (note that order
relation is not defined in the set of complex numbers)
∴ Roots of equation are either real or complex conjugate.
Let $$\alpha {\text{,}}\beta $$ be the roots of $$a{x^2} + bx + c = 0,$$ then
$$\alpha + \beta = - \frac{b}{a} = - ve,\,\,\,\,\,\,\alpha \beta = \frac{c}{a} = + ve$$
⇒ Either both $$\alpha {\text{,}}\beta $$ are $$- ve$$ (if roots are real) or both $$\alpha {\text{,}}\beta $$ have $$- ve$$ real parts (if roots are complex conjugate)