If - and - are two real roots of the equation x3-p x2-q x-r-0 satisfying the relation -1-0- then find the value of r2-p r-q

Solution: For nbsp; x^3 + p x^2 + qx + r = 0 Sum of the roots nbsp;α + nbsp;β + nbsp;γ = p _______(1) αβ + βγ + nbsp;γα nbsp;= q ________(2) αβγ = ...

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Standard XI

Mathematics

Geometric Progression

If α and β ar...

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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Solution

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

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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. __

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.

__