If α and β are two real roots of the equation x3 - p x2 + qx + r = 0 satisfying the relation αβ + 1 = 0, then find the value of r2 + pr + q.
Solution: For x3 + p x2 + qx + r = 0
Sum of the roots α + β + γ = -p _______(1)
αβ + βγ + γα = q ________(2)
αβγ = -r _________(3) {Given αβ = -1}
From equation 3 (-1)γ = -r
γ = r
Substituting γ in equation 2
-1 + (α + β)r = q
α + β = q+1r _________(4)
Substitute α + β+ γ = -p
q+1r + r = -p
q + 1 + r2 = -pr
r2 + pr + q + 1 = 0
r2 + pr + q = -1