If-root-of-equation-x-3-px-2-qx-r-0-are-in-AP-than-what-is-the-relation-between-p-q-and-r-

Question Number 106138 by bobhans last updated on 03/Aug/20

If root of equation x^{3} - {px}^{2} + qx - r = 0 are in

AP than what is the relation between p, q

and r ?

Answered by bemath last updated on 03/Aug/20

let x_{1} - d, x_{1}; x_{1} + d is the roots of the given equation . we have

\Rightarrow (x_{1} - d) + x_{1} + (x_{1} + d) = p [{Vietha}^{'} s rule]

\Rightarrow {3 x}_{1} = p \to x_{1} = \frac{p}{3}, so we get

{(\frac{p}{3})}^{3} - p {(\frac{p}{3})}^{2} + q (\frac{p}{3}) - r = 0

\frac{p^{3}}{27} - \frac{p^{3}}{9} + \frac{pq}{3} - r = 0

- {2 p}^{3} + 9 pq - 27 r = 0

{27 p}^{3} - 9 pq + 27 r = 0

Commented by bemath last updated on 03/Aug/20

thank you sir

Commented by Rasheed.Sindhi last updated on 03/Aug/20

W h y h a v e y o u a s s u m e d o n e r o o t

r (t h e c o n t a n t t e r m o f t h e

e q u a t i o n) ?

Commented by Rasheed.Sindhi last updated on 03/Aug/20

N i c e!

Commented by bemath last updated on 03/Aug/20

oo sorry . i can edit

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