 # If the zeroes of the quadratic polynomial `
Question:

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

(a) c and a have opposite signs

(b) c and b have opposite signs

(c) c and a have same signs

(d) c and b have the same signs

Solution:

(c) The zeroes of the given quadratic polynomial ax2 + bx + c, c ≠ 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) $x^{2}+4 x+4=0$    (ii) $x^{2}-4 x+4=0$

$\Rightarrow \quad(x+2)^{2}=0 \quad \Rightarrow \quad(x-2)^{2}=0$

$\Rightarrow \quad x=-2,-2 \quad \Rightarrow \quad x=2,2$

Alternate Method

Given that, the zeroes of the quadratic polynomial $a x^{2}+b x+c$, where $c \neq 0$, are equal i.e., discriminant $(D)=0$

$\Rightarrow \quad b^{2}-4 a c=0$

$\Rightarrow \quad b^{2}=4 a c$

$\Rightarrow \quad a c=\frac{b^{2}}{4}$

$\Rightarrow \quad \quad a c>0$

which is only possible when a and c have the same signs.