If α and β are the roots of the equation x^2 - q(1+x) - r = 0, then what is (1+α)(1+β)

• Quadratic Equations

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?

1. 1 - r
2. q - r
3. 1 + r
4. q + r

Solution

x² - q - qx - r = 0

x² - qx - (q + r) = 0

Sum of roots = α + β = -b/a = q

Product of roots = αβ = c/a = -(q+r)

(1+α)(1+β) = 1 + (α+β) + αβ

= 1 + q - q - r 

= 1 - r

The correct option is A.