1/2 x (x+2) (2x+3) = 138^2

asked by guest
on Jan 13, 2025 at 1:08 am



You asked:

Solve the equation \(\frac{1}{2} \cdot x\left( x + 2 \right) \left(2 x + 3\right) = {138}^{2}\) for the variable \(x\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= - \frac{7}{6} + \sqrt[3]{\frac{2056787}{216} + \frac{\sqrt{1450745117}}{4}} + \frac{13}{36 \sqrt[3]{\frac{2056787}{216} + \frac{\sqrt{1450745117}}{4}}} \approx 25.551606\\x &= - \frac{\sqrt[3]{\frac{\sqrt{1450745117}}{4} + \frac{2056787}{216}}}{2} - \frac{7}{6} - \frac{13}{72 \sqrt[3]{\frac{\sqrt{1450745117}}{4} + \frac{2056787}{216}}} + i \left(- \frac{13 \sqrt{3}}{72 \sqrt[3]{\frac{\sqrt{1450745117}}{4} + \frac{2056787}{216}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1450745117}}{4} + \frac{2056787}{216}}}{2}\right) \approx -14.525803 + 23.115281 i\\x &= - \frac{\sqrt[3]{\frac{\sqrt{1450745117}}{4} + \frac{2056787}{216}}}{2} - \frac{7}{6} - \frac{13}{72 \sqrt[3]{\frac{\sqrt{1450745117}}{4} + \frac{2056787}{216}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1450745117}}{4} + \frac{2056787}{216}}}{2} + \frac{13 \sqrt{3}}{72 \sqrt[3]{\frac{\sqrt{1450745117}}{4} + \frac{2056787}{216}}}\right) \approx -14.525803 -23.115281 i\end{aligned}\]

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

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