Equal Zeroes of a Quadratic Polynomial

Video Explanation

Question

If the zeroes of the quadratic polynomial

\[ f(x) = ax^2 + bx + c, \quad c \neq 0 \]

are equal, then:

(a) \(c\) and \(a\) have opposite signs
(b) \(c\) and \(b\) have opposite signs
(c) \(c\) and \(a\) have the same sign
(d) \(c\) and \(b\) have the same sign

Solution

Step 1: Use the Condition for Equal Zeroes

For a quadratic polynomial, the zeroes are equal if the discriminant is zero.

\[ \text{Discriminant} = b^2 – 4ac \]

So,

\[ b^2 – 4ac = 0 \]

Step 2: Analyze the Sign of \(ac\)

From the above equation:

\[ 4ac = b^2 \]

Since \(b^2 > 0\), we get:

\[ ac > 0 \]

This means that \(a\) and \(c\) must have the same sign.

Conclusion

The correct answer is:

\[ \boxed{\text{(c) } c \text{ and } a \text{ have the same sign}} \]