Question

If the difference of the roots of $x^2-px+q=0$ is unity, then
(a) $p^2+4q=1$
(b) $p^2-4q=1$
(c) $p^2+4q^2=(1+2q{)}^2$
(d) $4p^2+q^2=(1+2p{)}^2$

Answer 1

(b) $p^2-4q=1$
Given equation: $x^2-px+q=0$
Also $\alpha and\beta$ are the roots of the equation such that $\alpha -\beta =1$.
Sum of the roots = $\alpha +\beta =\frac{-\mathrm{Coefficient}\mathrm{of}x}{\mathrm{Coefficient}\mathrm{of}{x}^2}=-\left(\frac{-p}{1}\right)=p$

Product of the roots = $\alpha \beta =\frac{C\mathrm{onstant}\mathrm{term}}{C\mathrm{oefficient}\mathrm{of}{x}^2}=q$

$$\begin{gathered} \therefore (\alpha +\beta {)}^2-(\alpha -\beta {)}^2=4\alpha \beta \\ \Rightarrow p^2-1=4q \\ \Rightarrow p^2-4q=1 \end{gathered}$$