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You asked:
Solve the equation
x 3 − 5 x 2 + 3 x − 9 = 0 {x}^{3} - 5 {x}^{2} + 3 x - 9 = 0 x 3 − 5 x 2 + 3 x − 9 = 0 for the variable
x x x .
MathBot Answer:
The 3 solutions to the equation are:
x = 5 3 + 179 27 + 345 3 3 + 16 9 179 27 + 345 3 3 ≈ 4.7667344 x = − 345 3 + 179 27 3 2 − 8 9 345 3 + 179 27 3 + 5 3 + i ( − 8 3 9 345 3 + 179 27 3 + 3 345 3 + 179 27 3 2 ) ≈ 0.11663278 + 1.3691172 i x = − 345 3 + 179 27 3 2 − 8 9 345 3 + 179 27 3 + 5 3 + i ( − 3 345 3 + 179 27 3 2 + 8 3 9 345 3 + 179 27 3 ) ≈ 0.11663278 − 1.3691172 i \begin{aligned}x &= \frac{5}{3} + \sqrt[3]{\frac{179}{27} + \frac{\sqrt{345}}{3}} + \frac{16}{9 \sqrt[3]{\frac{179}{27} + \frac{\sqrt{345}}{3}}} \approx 4.7667344\\x &= - \frac{\sqrt[3]{\frac{\sqrt{345}}{3} + \frac{179}{27}}}{2} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{345}}{3} + \frac{179}{27}}} + \frac{5}{3} + i \left(- \frac{8 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{345}}{3} + \frac{179}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{345}}{3} + \frac{179}{27}}}{2}\right) \approx 0.11663278 + 1.3691172 i\\x &= - \frac{\sqrt[3]{\frac{\sqrt{345}}{3} + \frac{179}{27}}}{2} - \frac{8}{9 \sqrt[3]{\frac{\sqrt{345}}{3} + \frac{179}{27}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{345}}{3} + \frac{179}{27}}}{2} + \frac{8 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{345}}{3} + \frac{179}{27}}}\right) \approx 0.11663278 -1.3691172 i\end{aligned} x x x = 3 5 + 3 27 179 + 3 345 + 9 3 27 179 + 3 345 16 ≈ 4.7667344 = − 2 3 3 345 + 27 179 − 9 3 3 345 + 27 179 8 + 3 5 + i − 9 3 3 345 + 27 179 8 3 + 2 3 3 3 345 + 27 179 ≈ 0.11663278 + 1.3691172 i = − 2 3 3 345 + 27 179 − 9 3 3 345 + 27 179 8 + 3 5 + i − 2 3 3 3 345 + 27 179 + 9 3 3 345 + 27 179 8 3 ≈ 0.11663278 − 1.3691172 i
i i i imaginary unit , defined as i 2 = − 1 i^2 = -1 i 2 = − 1
