Nature of Zeroes of a Quadratic Polynomial

Video Explanation

Question

The zeroes of the quadratic polynomial

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

are:

(a) cannot both be positive
(b) cannot both be negative
(c) are always unequal
(d) are always equal

Solution

Step 1: Use Relations Between Zeroes and Coefficients

For a quadratic polynomial \(x^2 + ax + a\),

Sum of zeroes:

\[ \alpha + \beta = -a \]

Product of zeroes:

\[ \alpha\beta = a \]

Step 2: Check Possibility of Both Zeroes Being Positive

If both zeroes are positive, then:

\[ \alpha + \beta > 0 \quad \text{and} \quad \alpha\beta > 0 \]

From above,

\[ -a > 0 \Rightarrow a < 0 \]

and

\[ a > 0 \]

This is a contradiction.

Step 3: Conclusion from the Contradiction

Hence, the zeroes of the polynomial cannot both be positive.

Final Answer

The correct option is:

\[ \boxed{\text{(a) cannot both be positive}} \]