x^3+3x^2-20x+43=0

asked by guest
on Jan 24, 2025 at 12:20 am



You asked:

Solve the equation \({x}^{3} + 3 {x}^{2} - 20 x + 43 = 0\) for the variable \(x\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= -1 - \frac{23}{\sqrt[3]{\frac{1755}{2} + \frac{3 \sqrt{196221}}{2}}} - \frac{\sqrt[3]{\frac{1755}{2} + \frac{3 \sqrt{196221}}{2}}}{3} \approx -6.8418059\\x &= -1 + \frac{23}{2 \sqrt[3]{\frac{3 \sqrt{196221}}{2} + \frac{1755}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{196221}}{2} + \frac{1755}{2}}}{6} + i \left(- \frac{23 \sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{196221}}{2} + \frac{1755}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{196221}}{2} + \frac{1755}{2}}}{6}\right) \approx 1.920903 + 1.6109073 i\\x &= -1 + \frac{23}{2 \sqrt[3]{\frac{3 \sqrt{196221}}{2} + \frac{1755}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{196221}}{2} + \frac{1755}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{196221}}{2} + \frac{1755}{2}}}{6} + \frac{23 \sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{196221}}{2} + \frac{1755}{2}}}\right) \approx 1.920903 -1.6109073 i\end{aligned}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

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