x^2 +x^3= 5

asked by guest
on Sep 20, 2024 at 4:31 am



You asked:

Solve the equation \({x}^{2} + {x}^{3} = 5\) for the variable \(x\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= - \frac{1}{3} + \sqrt[3]{\frac{133}{54} + \frac{\sqrt{1965}}{18}} + \frac{1}{9 \sqrt[3]{\frac{133}{54} + \frac{\sqrt{1965}}{18}}} \approx 1.4334277\\x &= - \frac{\sqrt[3]{\frac{\sqrt{1965}}{18} + \frac{133}{54}}}{2} - \frac{1}{3} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{1965}}{18} + \frac{133}{54}}} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{1965}}{18} + \frac{133}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1965}}{18} + \frac{133}{54}}}{2}\right) \approx -1.2167138 + 1.4169509 i\\x &= - \frac{\sqrt[3]{\frac{\sqrt{1965}}{18} + \frac{133}{54}}}{2} - \frac{1}{3} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{1965}}{18} + \frac{133}{54}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1965}}{18} + \frac{133}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{1965}}{18} + \frac{133}{54}}}\right) \approx -1.2167138 -1.4169509 i\end{aligned}\]

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

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