2x$x^{2}$+$x$+$\frac14$=30+$\frac14$

asked by guest
on Dec 04, 2024 at 10:29 am



You asked:

Solve the equation \(2 x \cdot {x}^{2} + x + \frac{1}{4} = 30 + \frac{1}{4}\) for the variable \(x\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= \sqrt[3]{\frac{15}{2} + \frac{\sqrt{72906}}{36}} - \frac{1}{6 \sqrt[3]{\frac{15}{2} + \frac{\sqrt{72906}}{36}}} \approx 2.3986494\\x &= - \frac{\sqrt[3]{\frac{15}{2} + \frac{\sqrt{72906}}{36}}}{2} + \frac{1}{12 \sqrt[3]{\frac{15}{2} + \frac{\sqrt{72906}}{36}}} + i \left(\frac{\sqrt{3}}{12 \sqrt[3]{\frac{15}{2} + \frac{\sqrt{72906}}{36}}} + \frac{\sqrt{3} \sqrt[3]{\frac{15}{2} + \frac{\sqrt{72906}}{36}}}{2}\right) \approx -1.1993247 + 2.1943426 i\\x &= - \frac{\sqrt[3]{\frac{15}{2} + \frac{\sqrt{72906}}{36}}}{2} + \frac{1}{12 \sqrt[3]{\frac{15}{2} + \frac{\sqrt{72906}}{36}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{15}{2} + \frac{\sqrt{72906}}{36}}}{2} - \frac{\sqrt{3}}{12 \sqrt[3]{\frac{15}{2} + \frac{\sqrt{72906}}{36}}}\right) \approx -1.1993247 -2.1943426 i\end{aligned}\]

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

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