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brotherbobby

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- Homework Statement
- At what values of ##a## does the equation ##2x^2-(a^3+8a-1)x+a^2-4a = 0## possess roots of opposite signs?

- Relevant Equations
- For a quadratic equation ##ax^2+bx+c=0## having roots ##\alpha,\beta##, the sum of the roots ##\alpha+\beta = -\frac{b}{a}## and product of the roots ##\alpha\beta = \frac{c}{a}##.

**Given :**The equation ##2x^2-(a^3+8a-1)x+a^2-4a = 0## with roots of opposite signs.

**Required :**What is the value of ##a## ?

**Attempt :**The roots of the equation must be of the form ##\alpha, -\alpha##. The sum of the roots ##0 = a^3+8a-1##.

I do not know how to solve this equation.

However, this is not the answer in the book.

**Answer :**##a \in (0;4)## (from book)

Any help would be welcome.