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Question
If the difference of the roots of \[x^2 - px + q = 0\] is unity, then
Options
\[p^2 + 4q = 1\]
\[p^2 - 4q = 1\]
\[p^2 + 4 q^2 = (1 + 2q )^2\]
\[4 p^2 + q^2 = (1 + 2p )^2\]
Solution
\[p^2 - 4q = 1\]
Given equation:
\[x^2 - px + q = 0\]
Also
\[\alpha \text { and } \beta\] are the roots of the equation such that \[\alpha - \beta = 1\].
Sum of the roots = \[\alpha + \beta = \frac{- \text { Coefficient of } x}{\text { Coefficient of } x^2} = - \left( \frac{- p}{1} \right) = p\]
Product of the roots = \[\alpha\beta = \frac{\text { Constant term }}{\text { Coefficient of } x^2} = q\]
\[\therefore (\alpha + \beta )^2 - (\alpha - \beta )^2 = 4\alpha\beta\]
\[ \Rightarrow p^2 - 1 = 4q\]
\[ \Rightarrow p^2 - 4q = 1\]
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