Find a Quadratic Polynomial Whose Sum and Product of Zeros Are −3/(2√5) and −1/2 and Find Its Zeros by Factorisation

Video Explanation

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Solution

Given:

Sum of zeros = −3/(2√5)
Product of zeros = −1/2

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 − (−3/(2√5))x − 1/2

x² + (3/(2√5))x − 1/2

Multiplying throughout by 2√5 to remove fractions:

2√5x² + 3x − √5

Step 2: Find the Zeros by Factorisation

2√5x² + 3x − √5 = 0

Factorising:

(√5x − 1)(2x + √5) = 0

∴ √5x − 1 = 0   or   2x + √5 = 0

∴ x = 1/√5   or   x = −√5/2

Zeros of the polynomial are 1/√5 and −√5/2.

Step 3: Verification

Let α = 1/√5 and β = −√5/2

Sum of zeros:

α + β = 1/√5 − √5/2

= (2 − 5)/(2√5) = −3/(2√5)

✔ Verified

Product of zeros:

αβ = (1/√5)(−√5/2) = −1/2

✔ Verified

Final Answer

Required quadratic polynomial is 2√5x² + 3x − √5.

Zeros of the polynomial are 1/√5 and −√5/2.

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.