Question

If the zeros of the quadratic polynomial ax² + bx + c, where a ≠ 0 and c ≠ 0, are equal then

(a) c and a have the same sign
(b) c and a have opposite signs
(c) c and b have the same sign
(d) c and b have opposite sign

Answer 1

$$\begin{gathered} \left(\text{a}\right)\mathit{\text{c}}\text{and}\mathit{\text{a}}\text{have the same sign} \\ \text{Let}\alpha \text{and}\alpha \text{be the zeroes of}a{x}^2+bx+c. \\ \text{Then}\alpha +\alpha =\frac{-b}{a} \\ >2\alpha =\frac{-b}{a} \\ =>\alpha =\frac{-b}{2a} \\ \text{Also,}\alpha \times \alpha =\frac{c}{a} \\ =>{\alpha }^2=\frac{c}{a}>0 \\ \therefore c\text{and}a\text{must have the same sign}\text{.} \end{gathered}$$