if-a-and-b-are-the-root-of-the-quadratic-equation-x-2-2x-3-then-find-the-value-of-2-2-

Question Number 25184 by lucky singh last updated on 05/Dec/17

i f a a n d b a r e t h e r o o t o f t h e q u a d r a t i c e q u a t i o n x^{2} + 2 x + 3 t h e n f i n d t h e v a l u e o f α^{2} / β + β^{2} / α

Commented by prakash jain last updated on 05/Dec/17

x^{2} + 2 x + 3 = 0

α + β = - 2

α β = 3

\frac{α^{2}}{β} + \frac{β^{2}}{α}

= \frac{α^{3} + β^{3}}{α β} = \frac{(α + β) (α^{2} + β^{2} - α β)}{α β}

= \frac{(α + β) ({(α + β)}^{2} - 3 α β)}{α β}

= \frac{- 2 ({(- 2)}^{2} - 3 \times 3)}{3}

= \frac{- 2 (- 5)}{3} = \frac{10}{3}

Answered by ibraheem160 last updated on 06/Dec/17

α + β = - 2

α β = 3

\frac{α^{2}}{β} + \frac{β^{2}}{α} = \frac{α^{3} + β^{3}}{α β}

α^{3} + β^{3} = {(α + β)}^{3} - 3 α β (α + β)

\Rightarrow {(- 2)}^{3} - 3 (3) (- 2)

= 10

∴ \frac{α^{3} + β^{3}}{α β} = \frac{10}{3}