Question

if roots of the cubic equation x^3 − 7x^2+cx − 8=0 are in g.p. then the value of c is​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{Roots of the cubic eqution}\;\mathsf{x^3-7\,x^2+cx-8=0\;are\;in\;G.P}$

$\underline{\textbf{To find:}}$

$\textsf{The value of 'c'}$

$\underline{\textbf{Solution:}}$

$\mathsf{Let\;the\;roots\;be\;\dfrac{a}{r},a,a\,r}$

$\mathsf{Product\;of\;roots=\dfrac{-(-8)}{1}}$

$\mathsf{\dfrac{a}{r}{\times}a{\times}a\,r=8}$

$\mathsf{a^3=8}$

$\mathsf{a=2}$

$\implies\textsf{2 is one of the root of the equation}$

$\textsf{So, the equation will be satisfied by 2}$

$\implies\mathsf{(2)^3-7\,(2)^2+c(2)-8=0}$

$\implies\mathsf{8-28+2\,c-8=0}$

$\implies\mathsf{-28+2\,c=0}$

$\implies\mathsf{2\,c=28}$

$\implies\boxed{\mathsf{c=14}}$

$\underline{\textbf{Answer:}}$

$\textsf{The value of c is 14}$

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Answer 2

Roots of the cubic equation x^3 − 7x^2+cx − 8=0 are in g.p.    (given)

Here we have to find out the value of 'C'

While solving cubic equation, we will get three roots.

As the roots are in geometric progression,

Let consider the roots can are :-

$\frac{p}{q}$ ' $p$, $pq$            

product of the roots = - ( -8) = 8

∴ $\frac{p}{q}$ × $p$ × $pq$ =8

p³ = 8 = 2³

p = 2

Let us put 2 in the equation as a root.

x³ − 7x²+cx − 8=0

( 2 )³ - 7 ( 2)² + 2c - 8 =0

8- 28 + 2c -8=02c = 28

c = 14

Hence value of the 'C' will be 14.

Ans :- The value of ' C ' will be 14.

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