Question

If $\frac{x+1}{x^2-px+q}=\frac{A}{x-\alpha }+\frac{B}{x-\beta }$, where $\alpha$ and $\beta$ are the roots of $x^2-px+q=0$, then $\frac{A-B}{A+B}=\frac{p+2}{\sqrt{p^2-4q}}$ . If this is true enter 1, else enter 0.

Answer 1

$\frac{x+1}{x^2-px+q}=\frac{A}{x-\alpha }+\frac{B}{x-\beta }\Rightarrow x+1=A(x-\beta )+B(x-\alpha )\Rightarrow A+B=1,-\beta A-\alpha B=1$
As $\alpha ,\beta$ are roots of $x^2-px+q=0$
Then $\alpha +\beta =p$ and $\alpha \beta =q$
Hence $\frac{A-B}{A+B}=\frac{p+2}{\sqrt{p^2-4q}}$