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Carleigh Young

Answered question

2022-03-12

What should be the relationship between p and q, so that

x^{4} + p x^{2} + q = 0

equations has four solutions which are successive members of arithmetic equation.

rywleob8

Beginner2022-03-13Added 4 answers

Step 1
Ignoring the case

p = q = 0

, the solutions to this equation will be of the form

x = \pm α, \pm β (α > β > 0)

and the arithmetic sequence (in order) would be

- α, - β, β, α

By definition of an arithmetic progression,

α - β = 2 β \Rightarrow α = 3 β

Applying Vieta’s formulas on

x^{4} + p x^{2} + q = 0

α^{2} + β^{2} = - p

α^{2} β^{2} = q

\Rightarrow 10 β^{2} = - p

3 β^{2} = \sqrt{q}

\frac{\sqrt{q}}{p} = \frac{- 3}{10}

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