0=$2x^{5}$- $5x^{4}$+ $2x^{3}$+ $2x^{2}$ +x-1

asked by guest
on Jan 24, 2025 at 4:55 pm



You asked:

Solve the equation \(0 = 2 {x}^{5} - 5 {x}^{4} + 2 {x}^{3} + 2 {x}^{2} + x - 1\) for the variable \(x\).

MathBot Answer:

The 5 solutions to the equation are: \[\begin{aligned}x &= \frac{1}{2} = 0.5\\x &= \frac{1}{2} + \frac{\sqrt[4]{21} \cos{\left(\frac{\arctan{\left(\frac{2 \sqrt{3}}{3} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{21} i \sin{\left(\frac{\arctan{\left(\frac{2 \sqrt{3}}{3} \right)}}{2} \right)}}{2} \approx 1.4735615 -0.44477181 i\\x &= \frac{1}{2} + \frac{\sqrt[4]{21} \cos{\left(\frac{\arctan{\left(\frac{2 \sqrt{3}}{3} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{21} i \sin{\left(\frac{\arctan{\left(\frac{2 \sqrt{3}}{3} \right)}}{2} \right)}}{2} \approx 1.4735615 + 0.44477181 i\\x &= - \frac{\sqrt[4]{21} \cos{\left(\frac{\arctan{\left(\frac{2 \sqrt{3}}{3} \right)}}{2} \right)}}{2} + \frac{1}{2} - \frac{\sqrt[4]{21} i \sin{\left(\frac{\arctan{\left(\frac{2 \sqrt{3}}{3} \right)}}{2} \right)}}{2} \approx -0.47356148 -0.44477181 i\\x &= - \frac{\sqrt[4]{21} \cos{\left(\frac{\arctan{\left(\frac{2 \sqrt{3}}{3} \right)}}{2} \right)}}{2} + \frac{1}{2} + \frac{\sqrt[4]{21} i \sin{\left(\frac{\arctan{\left(\frac{2 \sqrt{3}}{3} \right)}}{2} \right)}}{2} \approx -0.47356148 + 0.44477181 i\end{aligned}\]

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

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