1/2{x-1/3[x-1/4(x-2/3)]-2/3}=x+3/4, find x

asked by guest
on Nov 23, 2024 at 8:02 pm



You asked:

Solve the equation \[\frac{1}{2 \cdot \left(x - \frac{1}{3 \cdot \left(x - \frac{1}{4 \cdot \left(x - \frac{2}{3}\right)}\right)} - \frac{2}{3}\right)} = x + \frac{3}{4}\] for the unknown \(x\).

MathBot Answer:

The complex solutions are: \[\left\{\frac{7}{48} - \frac{\sqrt{\frac{2035}{1728} + \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}} + 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}}{2} - \frac{\sqrt{\frac{2035}{864} - 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}} + \frac{59}{256 \sqrt{\frac{2035}{1728} + \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}} + 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}} - \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}}}{2}, \frac{7}{48} - \frac{\sqrt{\frac{2035}{1728} + \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}} + 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}}{2} + \frac{\sqrt{\frac{2035}{864} - 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}} + \frac{59}{256 \sqrt{\frac{2035}{1728} + \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}} + 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}} - \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}}}{2}, \frac{7}{48} - \frac{\sqrt{\frac{2035}{864} - 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}} - \frac{59}{256 \sqrt{\frac{2035}{1728} + \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}} + 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}} - \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}}}{2} + \frac{\sqrt{\frac{2035}{1728} + \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}} + 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}}{2}, \frac{7}{48} + \frac{\sqrt{\frac{2035}{864} - 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}} - \frac{59}{256 \sqrt{\frac{2035}{1728} + \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}} + 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}} - \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}}}{2} + \frac{\sqrt{\frac{2035}{1728} + \frac{45451}{93312 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}} + 2 \sqrt[3]{\frac{1147429}{10077696} + \frac{\sqrt{13210430763} i}{2985984}}}}{2}\right\} \setminus \left\{- \frac{1}{2}, \frac{2}{3}, \frac{7}{6}\right\}\]

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

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