If a, b, c > 0, then both the roots of the equation ax2 + bx + c = 0
-
are real and negative
-
have negative real part
-
are rational
-
are always imaginary
For ax^2 + bx + c = 0 with a, b, c > 0, the roots are given by (-b +/- sqrt(b^2 - 4ac)) / 2a. The real part is -b/2a, which is negative. If the discriminant is negative, the roots are complex with a negative real part; if positive, the roots are negative real numbers.
Using the quadratic formula, the roots of the equation are given by x = (-b + sqrt(b^2 - 4ac)) / 2a and x = (-b - sqrt(b^2 - 4ac)) / 2a. Because a, b, and c are positive, the sum of the roots (-b/a) is negative and the product of the roots (c/a) is positive. The product being positive means both roots have the same sign, and the sum being negative forces that sign to be negative. This holds true whether the discriminant yields real numbers or complex conjugates, so the roots will always have a negative real part.