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Question
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.
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Solution
Given
α + β = 24 ..............(i)
α − β = 8 ..............(ii)
By subtracting equation (ii) from (i) we get
α + β = 24
α − β = 8
--------------
2α = 32
`alpha=32/2`
α = 16
Substituting α = 16 in equation (i) we get,
α + β = 24
16 + β = 24
β = 24 - 16
β = 8
Let S and P denote respectively the sum and product of zeros of the required polynomial. then,
S = α + β
= 16 + 8
= 24
P = αβ
= 16 x 8
= 128
Hence, the required polynomial if f(x) is given by
f(x) = k(x2 - Sx + P)
f(x) = k(x2 -24x + 128)
Hence, required equation is f(x) = k(x2 -24x + 128) where k is any non-zeros real number.
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