3/16(x-1)-5/24(x-4)=2/5(x-6)+3/8

asked by guest
on Nov 16, 2024 at 4:29 am



You asked:

Solve the equation \(\frac{3}{16 \cdot \left(x - 1\right)} - \frac{5}{24 \cdot \left(x - 4\right)} = \frac{2}{5 \cdot \left(x - 6\right)} + \frac{3}{8}\) for the variable \(x\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= \frac{889}{270} + \frac{\sqrt{66721} \cos{\left(\frac{\arctan{\left(\frac{540 \sqrt{7190427}}{17173369} \right)}}{3} \right)}}{135} \approx 5.2052041\\x &= - \frac{\sqrt{66721} \cos{\left(\frac{\arctan{\left(\frac{540 \sqrt{7190427}}{17173369} \right)}}{3} \right)}}{540} + \frac{66721 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{17173369}{19683000} + \frac{\sqrt{7190427} i}{36450}}}\right)}}{72900} + \frac{\sqrt{200163} \sin{\left(\frac{\arctan{\left(\frac{540 \sqrt{7190427}}{17173369} \right)}}{3} \right)}}{540} + \frac{889}{270} + i \left(- \frac{\sqrt{200163} \cos{\left(\frac{\arctan{\left(\frac{540 \sqrt{7190427}}{17173369} \right)}}{3} \right)}}{540} - \frac{\sqrt{66721} \sin{\left(\frac{\arctan{\left(\frac{540 \sqrt{7190427}}{17173369} \right)}}{3} \right)}}{540} + \frac{66721 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{17173369}{19683000} + \frac{\sqrt{7190427} i}{36450}}}\right)}}{72900}\right) \approx 2.3827426 + 1.0 \cdot 10^{-143} i\\x &= \frac{66721 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{17173369}{19683000} + \frac{\sqrt{7190427} i}{36450}}}\right)}}{72900} - \frac{\sqrt{66721} \cos{\left(\frac{\arctan{\left(\frac{540 \sqrt{7190427}}{17173369} \right)}}{3} \right)}}{540} - \frac{\sqrt{200163} \sin{\left(\frac{\arctan{\left(\frac{540 \sqrt{7190427}}{17173369} \right)}}{3} \right)}}{540} + \frac{889}{270} + i \left(\frac{66721 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{17173369}{19683000} + \frac{\sqrt{7190427} i}{36450}}}\right)}}{72900} - \frac{\sqrt{66721} \sin{\left(\frac{\arctan{\left(\frac{540 \sqrt{7190427}}{17173369} \right)}}{3} \right)}}{540} + \frac{\sqrt{200163} \cos{\left(\frac{\arctan{\left(\frac{540 \sqrt{7190427}}{17173369} \right)}}{3} \right)}}{540}\right) \approx 2.2898311 -2.0 \cdot 10^{-145} i\end{aligned}\]

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

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