Question: obtaining real roots of a cubic polynomium

I have a matrix A4  


When calculting the Eigenvalues I get, as expected,


Then, I would like to calculate the roots, which are real. So, I replace

and define F as

I assume that z is real

and calculate the root by

My problem is that roots 2 and 3 are not real. I need to find a solution only when all the roots are real. How can I do that?

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