If the zeroes of the quadratic polynomial ,ax bx c2 + + 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 sign(d) c and b have the same sign

To solve this question, we need to consider the given quadratic polynomial, ax^2 + bx + c, where c is not equal to 0.

Step 1: Let's assume that the zeroes of the quadratic polynomial are equal. This means that the quadratic equation, ax^2 + bx + c = 0, will have the same value for both roots.

Step 2: Using the quadratic formula, we can find the roots of the quadratic equation. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

Step 3: Since the zeroes are equal, the discriminant (b^2 - 4ac) must be equal to zero. This is because when the discriminant is zero, the quadratic equation will have only one root.

Step 4: Setting the discriminant equal to zero, we have:

b^2 - 4ac = 0

Step 5: Now, let's analyze the signs of the coefficients a, b, and c.

If a and c have opposite signs, then ac will be negative. In order for the discriminant to be zero, b^2 must be equal to 4ac. Since ac is negative, b^2 must also be negative. This means that b will be imaginary, which contradicts the assumption that the zeroes are real.

If c and b have opposite signs, then bc will be negative. In order for the discriminant to be zero, b^2 must be equal to 4ac. Since bc is negative, b^2 must also be negative. This means that b will be imaginary, which again contradicts the assumption that the zeroes are real.

If c and a have the same sign, then ac will be positive. In order for the discriminant to be zero, b^2 must be equal to 4ac. Since ac is positive, b^2 must also be positive. This means that b will be real, which is consistent with the assumption that the zeroes are real.

Therefore, the correct answer is (c) c and a have the same sign.