Question 9

If the zeroes of the quadratic polynomial $a{x}^2+bx+c$, where $c\ne 0$, are equal, then

Solution:

The zeroes of the given quadratic polynomial $a{x}^2+bx+c,c\ne 0$ are equal. If coefficient of $x^2$ 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) ${\mathrm{x2}}^{}+4x+4=0\left(ii\right){\mathrm{x2}}^{}-4x+4=0$
$\Rightarrow {(x+2)2}^{}=0\Rightarrow {(x-2)2}^{}=0$
$\Rightarrow x=-2,-2x=2,2$

Alternative Method
Given that, the zeroes of the quadratic polynomial $a{\mathrm{x2}}^{}+bx+c$, where $\ne 0$, are equal i.e., discriminant $\left(D\right)=6$
$\Rightarrow {\mathrm{b2}}^{}-4ac=0$
$\Rightarrow {\mathrm{b2}}^{}=4ac$
$\Rightarrow ac=\frac{{\mathrm{b2/}}^{}}{4}$
$\Rightarrow ac>0$
Which is only possible when a and c have the same signs.