Question

If α and β are two real roots of the equation $x^3$ - p $x^2$ + qx + r = 0 satisfying the relation αβ + 1 = 0, then find the value of $r^2$ + pr + q.

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Answer 1

Solution: For $x^3$ + p $x^2$ + 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 = $\frac{q+1}{r}$ _________(4)

Substitute α + β+ γ = -p

$\frac{q+1}{r}$ + r = -p

q + 1 + $r^2$ = -pr

$r^2$ + pr + q + 1 = 0

$r^2$ + pr + q = -1