First slide

Theory of expressions

Question

The value of $λ,$ for which the sum of squares of the roots of the equation
$x^{2} - (λ + 2) x - λ + 1 = 0$ assumes the least value, is

Moderate

A

$3$
B

$\frac{1}{3}$
C

$- 3$
D

$- \frac{1}{3}$

Solution

The given equation is $x^{2} - (λ + 2) x - λ + 1 = 0$
If the roots of the given equation are $α, β$

Then Sum of the roots : $α + β = λ + 2$

and Product of the roots : $α β = 1 - λ$

by the given condition the sum of squares of the roots

Hence, $α^{2} + β^{2} = {(α + β)}^{2} - 2 α β$

                              $= {(λ + 2)}^{2} - 2 (1 - λ)$

                                $= λ^{2} + 4 λ + 4 - 2 + 2 λ$

                               $= λ^{2} + 6 λ + 9 - 7$

                               $= {(λ + 3)}^{2} - 7$

Clearly, $α^{2} + β^{2}$ is least value when $λ + 3 = 0$ i.e. When $λ = - 3.$

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