Find a Quadratic Polynomial Whose Sum and Product of Zeros Are −2√3 and −9 and Find Its Zeros by Factorisation

Video Explanation

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Solution

Given:

Sum of zeros = −2√3
Product of zeros = −9

Step 1: Form the Quadratic Polynomial

If sum of zeros = α + β and product of zeros = αβ, then the quadratic polynomial is:

x² − (α + β)x + αβ

∴ Required polynomial:

x² − (−2√3)x − 9

x² + 2√3x − 9

Step 2: Find the Zeros by Factorisation

x² + 2√3x − 9 = 0

Split the middle term:

x² + 3√3x − √3x − 9 = 0

Grouping the terms:

x(x + 3√3) − √3(x + 3√3) = 0

(x + 3√3)(x − √3) = 0

∴ x + 3√3 = 0   or   x − √3 = 0

∴ x = −3√3   or   x = √3

Zeros of the polynomial are −3√3 and √3.

Step 3: Verification

Let α = −3√3 and β = √3

Sum of zeros:

α + β = −3√3 + √3 = −2√3

✔ Verified

Product of zeros:

αβ = (−3√3)(√3) = −9

✔ Verified

Final Answer

Required quadratic polynomial is x² + 2√3x − 9.

Zeros of the polynomial are −3√3 and √3.

Conclusion

Thus, the quadratic polynomial formed using the given sum and product of zeros satisfies all the required conditions, and its zeros are obtained correctly by the factorisation method.