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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).
Solution
Given:
\[x^2 - a(x + 1) - c = 0 \text { or }x^2 - ax - a - c = 0\]
Also,
\[\alpha\] and \[\beta\] are the roots of the equation.
Sum of the roots = \[\alpha + \beta = - \left( \frac{- a}{1} \right) = a\]
Product of the roots = \[\alpha\beta = \frac{- (a + c)}{1} = - (a + c)\]
\[\therefore (1 + \alpha) (1 + \beta) = 1 + \beta + \alpha + \alpha\beta \]
\[ = 1 + (\alpha + \beta) + \left( \alpha\beta \right) \]
\[ = 1 + a - a - c \]
\[ = 1 - c\]
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