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[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

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...

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system

Let the equation

$$
x^{3}+a x^{2}+b x+c=0
$$

have roots \( x_{1}, x_{2}, x_{3} \). What relationship must hold between the coefficients of the equation in order for the following conditions to be true:

1. \( x_{2}=\frac{x_{1}+x_{3}}{2} \)
2. \( x_{2}=\sqrt{x_{1}x_{3}} \)
3. \( x_{2}=\frac{2x_{1}x_{3}}{x_{1}+x_{3}} \)

Determine the roots of the equation in each of the three cases.

Present the answer in LaTex format: \boxed{Your answer}
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