419 = (x+1)^2 *x

asked by guest
on Oct 26, 2024 at 9:25 pm



You asked:

Solve the equation \(419 = {\left( x + 1 \right)}^{2} \cdot x\) for the variable \(x\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= - \frac{2}{3} + \sqrt[3]{\frac{11315}{54} + \frac{\sqrt{14225469}}{18}} + \frac{1}{9 \sqrt[3]{\frac{11315}{54} + \frac{\sqrt{14225469}}{18}}} \approx 6.8315461\\x &= - \frac{\sqrt[3]{\frac{\sqrt{14225469}}{18} + \frac{11315}{54}}}{2} - \frac{2}{3} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{14225469}}{18} + \frac{11315}{54}}} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{14225469}}{18} + \frac{11315}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{14225469}}{18} + \frac{11315}{54}}}{2}\right) \approx -4.4157731 + 6.4679257 i\\x &= - \frac{\sqrt[3]{\frac{\sqrt{14225469}}{18} + \frac{11315}{54}}}{2} - \frac{2}{3} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{14225469}}{18} + \frac{11315}{54}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{14225469}}{18} + \frac{11315}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{14225469}}{18} + \frac{11315}{54}}}\right) \approx -4.4157731 -6.4679257 i\end{aligned}\]

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

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