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Existence of Limit

If α, β are r...

Question

If α, β are roots of the equation $x^{2} - a (x + 1) - c = 0$ , then write the value of (1 + α) (1 + β).

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Solution

Given: $x^{2} - a (x + 1) - c = 0 or x^{2} - a x - a - c = 0$
Also, $α$ and $β$ are the roots of the equation.
Sum of the roots = $α + β = - (\frac{- a}{1}) = a$

Product of the roots = $α β = \frac{- (a + c)}{1} = - (a + c)$

$∴ (1 + α) (1 + β) = 1 + β + α + α β = 1 + (α + β) + (α β) = 1 + a - a - c = 1 - c$

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