If α, β are the roots of the equation x

Question:

If $\alpha, \beta$ are the roots of the equation $x^{2}-p(x+1)-c=0$, then $(\alpha+1)(\beta+1)=$

(a) c

(b) c − 1

(c) 1 − c

(d) none of these

Solution:

(c) 1 − c

Given equation: $x^{2}-p(x+1)-c=0$

or $\quad x^{2}-p x-p-c=0$

Also $\alpha$ and $\beta$ are the roots of the equation.

Sum of the roots $=\alpha+\beta=p$

Product of the roots $=\alpha \beta=-(c+p)$

Then, $(\alpha+1)(\beta+1)=\alpha \beta+\alpha+\beta+1$

$=-(c+p)+p+1$

$=-c-p+p+1$

$=1-c$