find-the-relation-between-a-b-c-and-d-if-the-roots-of-the-eqution-ax-3-bx-2-cx-d-0-are-in-i-A-P-ii-G-P-

Question Number 9873 by j.masanja06@gmail.com last updated on 11/Jan/17

find the relation between a, b, c and d

if the roots of the eqution

{ax}^{3} + {bx}^{2} + cx + d = 0 are in;

(i) A . P

(ii) G . P

Answered by bansal22luvi@gmail.com last updated on 11/Jan/17

w h e n t h e r o o t s a r e i n A P

l e t t h e r o o t s b e (α - δ) . α . (α + δ)

3 α = - \frac{b}{a}

α = \frac{- b}{3 a} \dots \dots \dots \dots . . (1)

(α - δ) α (α + δ) = - \frac{d}{a}

α^{3} - α δ^{2} = - \frac{d}{a}

- \frac{b^{3}}{27 a^{3}} + \frac{b}{3 a} δ^{2} = - \frac{d}{a} (b y e q n 1)

\frac{- b^{3} + 9 a^{2} b δ^{2}}{27 a^{3}} = - \frac{d}{a}

δ^{2} = \frac{- 3 a^{2} d + b^{3}}{a^{2} b} \dots \dots . . (2)

α (α - δ) + (α - δ) (α + δ) + α (α + δ) = \frac{c}{a}

3 α^{2} - δ^{2} = \frac{c}{a}

\frac{b^{2}}{3 a^{2}} - \frac{b^{3} - 3 a^{2} d}{a^{2} b} = \frac{c}{a} (b y e q n 1 a n d 2)

\frac{- 2 b^{3} + 9 a^{2} d}{3 a^{2} b} = \frac{c}{a}

9 a^{2} d = 3 a b c + 2 b^{3}

s a m e a s d o n e w h e n t h e r o o t s a r e i n G P

K u s h