# If α, β are roots of the equation x

Question:

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

Solution:

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

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

Sum of the roots $=\alpha+\beta=-\left(\frac{-a}{1}\right)=a$

Product of the roots $=\alpha \beta=\frac{-(a+c)}{1}=-(a+c)$

$\therefore(1+\alpha)(1+\beta)=1+\beta+\alpha+\alpha \beta$

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

$=1+a-a-c$

$=1-c$