If the zeroes of the quadratic polynomial \(ax^2 + bx + c\), with \(c \ne 0\), are equal, then
\(c\) and \(a\) have opposite signs
\(c\) and \(b\) have opposite signs
\(c\) and \(a\) have the same sign
\(c\) and \(b\) have the same sign
Step 1: Condition for equal zeroes.
A quadratic equation \(ax^2 + bx + c = 0\) has equal roots when its discriminant is zero.
That is,
\(D = b^2 - 4ac = 0\).
So, \(b^2 = 4ac\).
Step 2: Interpret the relation.
Since \(b^2 \geq 0\), we see that \(4ac = b^2 \geq 0\).
This implies that \(ac \geq 0\).
Step 3: Exclude the zero case.
We are told \(c \ne 0\) and clearly \(a \ne 0\) (otherwise it is not quadratic).
So, \(ac \gt 0\).
Step 4: Final conclusion.
Thus, \(a\) and \(c\) must have the same sign.
Correct Option: (C)