Question

If the zeroes of the quadratic polynomial ax² + bx + c are both negative, then a, b and c all have the _______ sign.

Answer 1

Let f(x) = ax² + bx + c

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{zeroes}\mathrm{of}f\left(x\right)\mathrm{be}\alpha ,\beta ,\mathrm{where}\mathrm{all}\mathrm{these}\mathrm{zeroes}\mathrm{are}\mathrm{negative}. \\ \mathrm{Then}, \\ \alpha +\beta =-\frac{b}{a} \\ \because \alpha ,\beta \mathrm{are}\mathrm{negative} \\ \therefore \frac{b}{a}\mathrm{is}\mathrm{positive} \\ \Rightarrow \mathrm{either}\mathrm{both}a\mathrm{and}b\mathrm{are}\mathrm{positive}\mathrm{or}\mathrm{both}\mathrm{are}\mathrm{negative}. \\ \alpha \beta =\frac{c}{a} \\ \because \alpha ,\beta \mathrm{are}\mathrm{negative} \\ \therefore \frac{c}{a}\mathrm{is}\mathrm{positive} \\ \Rightarrow \mathrm{either}\mathrm{both}a\mathrm{and}c\mathrm{are}\mathrm{positive}\mathrm{or}\mathrm{both}\mathrm{are}\mathrm{negative}. \end{gathered}$$

Hence, if the zeroes of the quadratic polynomial ax² + bx + c are both negative, then a, b and c all have the same sign.