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The real numbers x_1, x_2, x_3 satisfying the equation x^3 - x

Question

The real numbers $x_{1}, x_{2}, x_{3}$ satisfying the equation $x^{3} - x^{2} + β x + γ = 0$ are in A.P.

All possible values of $γ$ are

(- \frac{1}{9}, \infty)

(- \frac{1}{27}, + \infty)

(\frac{2}{9}, + \infty)

none of these

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Eleanor Discussed

The real numbers

x_{1}, x_{2}, x_{3}

satisfying the equation

x^{3} - x^{2} + β x + γ = 0

are in A.P.

All possible values of

γ

are

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Text solutionVerified

Let

x_{1} = a - d, x_{2} = a, x_{3} = a + d

Sum of the roots of equations

x_{1} + x_{2} + x_{3} = 1

(By using properties of roots of equation

a - d + a + a + d = 1

3 a = 1

a = \frac{1}{3}

product of the roots of the equation

x_{1} x_{2} x_{3} = - γ

i.e

(a - d) (a) (a + d) = - γ

a d^{2} - a^{3} = γ

\frac{1}{3} (d^{2} - \frac{1}{9}) = γ

d^{2} \geq 0

so that

⟹

\frac{1}{3} (d^{2} - \frac{1}{9}) \geq \frac{- 1}{27}

γ ϵ [\frac{- 1}{27}, + \infty)

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Question Text	The real numbers $x_{1}, x_{2}, x_{3}$ satisfying the equation $x^{3} - x^{2} + β x + γ = 0$ are in A.P. All possible values of $γ$ are
Answer Type	Text solution:1
Upvotes	150