Question

If $\alpha$ and $\beta$ the roots of the equation $x^2-q(1+x)-r=0$ , then what is $(1+\alpha )(1+\beta )$ equal to ?

• A. 1 - r
• B. q - r
• C. 1 + r
• D. q + r

Answer 1

The correct option is A 1 - r

Given quadratic equation is $x^2-(1+x)q-r=0$

$\Rightarrow x^2-qx+(-q-r)=0$

Let $\alpha ,\beta$ be roots

Then $\alpha +\beta =\frac{-(-q)}{1}=q$

and $\alpha \beta =\frac{-q-r}{1}=-q-r$

The expression $(1+\alpha )(1+\beta )=1+\alpha \beta +\alpha +\beta =1+(-q-r)+\left(q\right)=1-r$