Question

Question 9
If the zeroes of the quadratic polynomial $ax2+bx+c,wherec\ne 0$ are equal then
(A) c and a have opposite signs
(B) c and b have opposite signs
(C) c and a have the same sign
(D) c and b have the same sign

Answer 1

The zeroes of the given quadratic polynomial $ax2+bx+c,c\ne 0$ are equal. if coefficient of x2 and constant term have the same sign i.e., c and a have the same sign. While b. i.e., coefficient of x can be positive/negative but not zero
e.g., (i) $x^2+4x+4=0$
$\Rightarrow (x+2{)}^2=0$
$\Rightarrow x=-3,-2$
(ii) $x^2–4x+4=0$
$\Rightarrow (x-2{)}^2=0$
$\Rightarrow x=2,2$
Alternate method
Given that, the zeroes of the quadratic polynomial $a{x}^2+bx+c$, where $c\ne 0$, are equal i.e., discriminate (D) =
$\Rightarrow b^2-4ac=0$
$\Rightarrow b^2=4ac$
$\Rightarrow ac=\frac{b^2}{4}$
$\Rightarrow ac>0$
Which is only possible when a and c have the same signs