Math, asked by jitesh00896, 5 months ago

find the relation between p q and R if the alpha beta and gamma are roots of the cubic equations x cube minus pX square + px minus r equals to zero that such that they are in AP​

Answers

Answered by shravanimore
1

Answer:

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Answered by kundan222025
0

Answer:

Given, x

3

−px

2

+qx−r=0

Case 1: Roots are α,−α,β

Sum of the roots, S

1

=α−α+β=p⟹β=p

Sum of the roots taken 2 at a time, S

2

=−α

2

+αβ−αβ=q⟹−α

2

=q

Product of the roots, $$S_3 = -\alpha^2\beta = r

From the above equations we can see that S

1

⋅S

3

=S

2

⟹pq=r

Case 2: Roots are in geometric progression. Let the roots be α,β,γ

β

α

=

γ

β

⟹β

2

=αγ⟹β

3

=αβγ=r

Since, β is the roots of the given equation, we have

β

3

−pβ

2

+qβ−r=0⟹r–p⋅r

2/3

+q⋅r

1/3

–r=0

⟹p

3

r=q

3

[taking cubes on both sides and simplifying]

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