Let $a > 0, b > 0,c >0 $ then both roots of the equation $ ax^2 + bx + c = 0$
-
are real and negative
-
have negative real parts
-
have positive real parts
-
none of these
For ax^2 + bx + c = 0 with a, b, c > 0, the roots are given by (-b +/- sqrt(b^2 - 4ac)) / 2a. Since a, b, c > 0, the real part -b/2a is negative. If roots are complex, they are conjugates with negative real parts. If real, they are negative because the sum of roots (-b/a) and product (c/a) are both positive, implying both roots must be negative.
Because all coefficients are strictly positive, the product of the roots c/a is positive and the sum of the roots -b/a is negative. This implies that if the roots are real, they must both be strictly negative. Furthermore, since the discriminant b^2 - 4ac could potentially be negative, the roots might be complex; in such a scenario, the complex roots would be conjugates with a real part equal to -b/(2a), which is also strictly negative. Therefore, regardless of whether the roots are real or complex, they must always have negative real parts.