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.

Answer 1

Given

α + β = 24         ..............(i)

α − β = 8           ..............(ii)

By subtracting equation (ii) from (i) we get

α + β = 24

α − β = 8

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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.