If α and β are the roots of the equation x² - q(1 + x) - r = 0, then what is (1 + α)(1 + β) equal to?

Concept:

Let us consider the standard form of a quadratic equation, 

ax² + bx + c =0

Let α and β be the two roots of the above quadratic equation. 

The sum of the roots of a quadratic equation is given by: \({\rm{α }} + {\rm{β }} = - \frac{{\rm{b}}}{{\rm{a}}} = - \frac{{{\rm{coefficient\;of\;x}}}}{{{\rm{coefficient\;of\;}}{{\rm{x}}^2}}}\) 

The product of the roots is given by:

 \({\rm{α β }} = \frac{{\rm{c}}}{{\rm{a}}} = \frac{{{\rm{constant\;term}}}}{{{\rm{coefficient\;of\;}}{{\rm{x}}^2}}}\)

Calculation:

Given:  α and β are the roots of the equation x² - q(1 + x) - r = 0

⇒ x² - q - qx - r = 0

⇒ x² - qx - (q + r) = 0

Sum of roots =  α + β = q

Product of roots = αβ = - (q + r) = -q - r

To find: (1 + α)(1 + β) 

(1 + α)(1 + β) = 1 + α + β + αβ 

= 1 + q - q - r 

= 1 - r