We must find the roots of the following cubic polynomial:

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Theory
The roots of the cubic equation:

are given by the following formulas:
![\begin{gathered} x_1=S+T-(1)/(3)a_1, \\ x_2=-(1)/(2)(S+T)-(1)/(3)a_1+(1)/(2)i\cdot\sqrt[]{3}\cdot(S-T), \\ x_3=-(1)/(2)(S+T)-(1)/(3)a_1-(1)/(2)i\cdot\sqrt[]{3}\cdot(S-T)\text{.} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/fclvwjkf273jv0ayo6p1nvibvy7cgb6tkk.png)
Where Q, S and T are given by:
![\begin{gathered} Q=(3a_2-a^2_1)/(9), \\ R=(9a_1a_2-27a_3-2a^3_1)/(54), \\ S=\sqrt[3]{R+\sqrt[]{Q^3+R^2}}, \\ T=\sqrt[3]{R-\sqrt[]{Q^3+R^2}}\text{.} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/bidonbgai695gy5g9ktiikl9ley13ymt4t.png)
The discriminant of the polynomial is:

We have the following possibilities:
0. one root is real and two complex conjugates if D > 0,
,
1. all roots are real and at last two are equal if D = 0,
,
2. all roots are real and unequal if D < 0.
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We rewrite the polynomial of the problem in the following way:

Comparing the last equation with the general case, we identify the coeffients:

Replacing the values of the coefficients in the formulas of Q and R, we get:

Replacing the values of Q and R in the formula of S and T, we get:
![\begin{gathered} S=\sqrt[3]{((9)/(2))+\sqrt[]{(-1)^3+((9)/(2))^2}}=\sqrt[]{(9)/(2)+\frac{\sqrt[]{77}}{2}}, \\ T=\sqrt[3]{((9)/(2))-\sqrt[]{(-1)^3+((9)/(2))^2}}=\sqrt[]{(9)/(2)-\frac{\sqrt[]{77}}{2}}\text{.} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/ty5ft6hnpizlbkhrwo0ze0mwyleksetcpv.png)
The discriminant of the polynomial is:
![D=Q^3+R^2=(-1)^3+(\frac{\sqrt[]{77}}{2})^2=(73)/(4)>0.^{}](https://img.qammunity.org/2023/formulas/mathematics/college/e0c4a9vghq3wdxo1fdudvukfmro3401of6.png)
We have D > 0 so one root is real and two complex conjugates.
We compute only the real root, which is given by:
![\begin{gathered} x_1=S+T-(1)/(3)a_1, \\ x_1=\sqrt[]{(9)/(2)+\frac{\sqrt[]{77}}{2}}+\sqrt[]{(9)/(2)-\frac{\sqrt[]{77}}{2}}-(1)/(3)(-3), \\ x_1=\sqrt[]{(9)/(2)+\frac{\sqrt[]{77}}{2}}+\sqrt[]{(9)/(2)-\frac{\sqrt[]{77}}{2}}+1, \\ x_1\cong3.55415. \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/ixavd6bnksr6o5cacfiitihu1ofec4ajm2.png)
Answer
The cubic equation has one real root and two complex roots. The real root is:
