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(C) For a quadratic polynomial $ax^{2} + bx + c$,the zeroes are equal if the discriminant $D = b^{2} - 4ac = 0$.This implies $b^{2} = 4ac$.Since $b^{2

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$c$ and $a$ have the same sign

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(C) For a quadratic polynomial $ax^{2} + bx + c$,the zeroes are equal if the discriminant $D = b^{2} - 4ac = 0$.This implies $b^{2} = 4ac$.Since $b^{2}$ is always non-negative for any real $b$,$4ac$ must be non-negative.Given $c 
eq 0$,if $a = 0$,it would not be a quadratic polynomial. Thus,$a 
eq 0$.Since $4ac = b^{2} \geq 0$,the product $ac$ must be greater than or equal to $0$.Since $c 
eq 0$ and $a 
eq 0$,$ac > 0$,which means $a$ and $c$ must have the same sign.For example:$(i)$ $x^{2} + 4x + 4 = 0 \Rightarrow (x+2)^{2} = 0 \Rightarrow x = -2, -2$ (Here $a=1, c=4$,both positive).$(ii)$ $x^{2} - 4x + 4 = 0 \Rightarrow (x-2)^{2} = 0 \Rightarrow x = 2, 2$ (Here $a=1, c=4$,both positive).

If the zeroes of the quadratic polynomial $ax^{2} + bx + c, c \neq 0$ are equal,then

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