The zeroes of the quadratic polynomial x² + ax + a (a ≠ 0)

Video Explanation

Watch the video explanation below:

Given

Quadratic polynomial: f(x) = x² + ax + a,   a ≠ 0

To Find

The correct statement about the zeroes of the given polynomial.

Solution

For a quadratic polynomial x² + ax + a:

Sum of zeroes = −a

Product of zeroes = a

Step 1: Analyse the Product of Zeroes

Product of zeroes = a

If both zeroes are positive or both negative, then their product must be positive.

So, this happens only when:

a > 0

Step 2: Analyse the Sum of Zeroes

Sum of zeroes = −a

If both zeroes are positive, then their sum must be positive.

But:

−a < 0 when a > 0

This is a contradiction.

Step 3: Draw the Conclusion

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

Final Answer

The correct statement is:

The zeroes cannot both be positive.

Correct Option

(a) cannot both be positive

Conclusion

Hence, for the quadratic polynomial x² + ax + a (a ≠ 0), the zeroes cannot both be positive.