Question

If the difference of the roots of \[x^2 - px + q = 0\]  is unity, then

 

विकल्प

• \[p^2 + 4q = 1\]

• \[p^2 - 4q = 1\]

• \[p^2 + 4 q^2 = (1 + 2q )^2\]

• \[4 p^2 + q^2 = (1 + 2p )^2\]

Answer 1

\[p^2 - 4q = 1\]

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 = \[\alpha\beta = \frac{\text { Constant term }}{\text { Coefficient of } x^2} = q\]

\[\therefore (\alpha + \beta )^2 - (\alpha - \beta )^2 = 4\alpha\beta\]

\[ \Rightarrow p^2 - 1 = 4q\]

\[ \Rightarrow p^2 - 4q = 1\]