If α and β are the zeros of a quadratic polynomial such that α + β = 24 and α − β = 8, find a quadratic polynomial having α and β as its zeros

Video Explanation

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Solution

Given:

α + β = 24
α − β = 8

Step 1: Find the Values of α and β

Adding the given equations:

(α + β) + (α − β) = 24 + 8

2α = 32

∴ α = 16

Subtracting the equations:

(α + β) − (α − β) = 24 − 8

2β = 16

∴ β = 8

Step 2: Form the Quadratic Polynomial

Sum of zeros = α + β = 24

Product of zeros = αβ = 16 × 8 = 128

The quadratic polynomial having zeros α and β is:

x² − (sum of zeros)x + (product of zeros)

∴ Required polynomial is:

x² − 24x + 128

Final Answer

The required quadratic polynomial is x² − 24x + 128.

Conclusion

Thus, the quadratic polynomial having zeros 16 and 8 is x² − 24x + 128.