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Question
If α and β are the zeros of the quadratic polynomial f(x) = x2 − p (x + 1) — c, show that (α + 1)(β +1) = 1− c.
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Solution
Since α and β are the zeros of the quadratic polynomial f(x) = x2 − p (x + 1) — c
Then
x2 - p(x + 1) - c
x2 - px - p - c
`alpha+beta="-coefficient of x"/("coefficient of "x^2)`
`=(-(-p))/1`
= p
`alphabeta="constant term"/("coefficient of "x^2)`
`=(-p-c)/1`
= -p-c
We have to prove that (α + 1)(β +1) = 1 − c
(α + 1)(β +1) = 1 - c
(α + 1)β + (α +1)(1) = 1 - c
αβ + β + α + 1 = 1 - c
αβ + (α + β) + 1 = 1 - c
Substituting α + β = p and αβ = -p-c we get,
-p - c + p + 1 = 1 - c
1 - c = 1 - c
Hence, it is shown that (α + 1)(β +1) = 1 - c
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