Question

If aplha and beta are the zeroes of the quadratic equation
$f{x} = {x}^{2} - p(x + 1) - c$
$show \: that \: ( \alpha + 1)( \beta + 1) = 1 - c$


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Answer 1

Solution :-

As given that α , β are the roots of the equation

f(x) = x² - p(x+1) - c = 0

Now by further solving f(x)

→ x² - px - p - c

→ x² - px - (p + c)

Now

Sum of roots

α + β = -b/a

α + β = p

Product of roots

αβ = c/a

αβ = -(p + c)

Now we will solve the equation

(α + 1) (β + 1)

→ αβ + (α + β) + 1

→ -(p + c) + (p) + 1

→ p - p + 1 - c

→ 1 - c

So Shown