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You asked:
Solve the equation
x 3 − x 2 − 8 x − 12 = 0 {x}^{3} - {x}^{2} - 8 x - 12 = 0 x 3 − x 2 − 8 x − 12 = 0 for the variable
x x x .
MathBot Answer:
The 3 solutions to the equation are:
x = 1 3 + 199 27 + 2 74 3 3 + 25 9 199 27 + 2 74 3 3 ≈ 3.8691878 x = − 2 74 3 + 199 27 3 2 − 25 18 2 74 3 + 199 27 3 + 1 3 + i ( − 25 3 18 2 74 3 + 199 27 3 + 3 2 74 3 + 199 27 3 2 ) ≈ − 1.4345939 + 1.0214531 i x = − 2 74 3 + 199 27 3 2 − 25 18 2 74 3 + 199 27 3 + 1 3 + i ( − 3 2 74 3 + 199 27 3 2 + 25 3 18 2 74 3 + 199 27 3 ) ≈ − 1.4345939 − 1.0214531 i \begin{aligned}x &= \frac{1}{3} + \sqrt[3]{\frac{199}{27} + \frac{2 \sqrt{74}}{3}} + \frac{25}{9 \sqrt[3]{\frac{199}{27} + \frac{2 \sqrt{74}}{3}}} \approx 3.8691878\\x &= - \frac{\sqrt[3]{\frac{2 \sqrt{74}}{3} + \frac{199}{27}}}{2} - \frac{25}{18 \sqrt[3]{\frac{2 \sqrt{74}}{3} + \frac{199}{27}}} + \frac{1}{3} + i \left(- \frac{25 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{74}}{3} + \frac{199}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{74}}{3} + \frac{199}{27}}}{2}\right) \approx -1.4345939 + 1.0214531 i\\x &= - \frac{\sqrt[3]{\frac{2 \sqrt{74}}{3} + \frac{199}{27}}}{2} - \frac{25}{18 \sqrt[3]{\frac{2 \sqrt{74}}{3} + \frac{199}{27}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{74}}{3} + \frac{199}{27}}}{2} + \frac{25 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{74}}{3} + \frac{199}{27}}}\right) \approx -1.4345939 -1.0214531 i\end{aligned} x x x = 3 1 + 3 27 199 + 3 2 74 + 9 3 27 199 + 3 2 74 25 ≈ 3.8691878 = − 2 3 3 2 74 + 27 199 − 18 3 3 2 74 + 27 199 25 + 3 1 + i − 18 3 3 2 74 + 27 199 25 3 + 2 3 3 3 2 74 + 27 199 ≈ − 1.4345939 + 1.0214531 i = − 2 3 3 2 74 + 27 199 − 18 3 3 2 74 + 27 199 25 + 3 1 + i − 2 3 3 3 2 74 + 27 199 + 18 3 3 2 74 + 27 199 25 3 ≈ − 1.4345939 − 1.0214531 i
i i i imaginary unit , defined as i 2 = − 1 i^2 = -1 i 2 = − 1
