(L^3)+9.362(L^2)-68.218=0

asked by guest
on Mar 17, 2025 at 2:35 am



You asked:

Solve the equation L3+9.362L268.218=0{L}^{3} + 9.362 {L}^{2} - 68.218 = 0 for the variable LL.

MathBot Answer:

The 3 solutions to the equation are: L=46811500+4681cos(arctan(1500460573988072930712548921759)3)7502.4075211L=468115004681cos(arctan(1500460573988072930712548921759)3)3000+21911761re(1(123i2)125489217593375000000+4605739880729307i22500003)2250000+46813sin(arctan(1500460573988072930712548921759)3)3000+i(46813cos(arctan(1500460573988072930712548921759)3)30004681sin(arctan(1500460573988072930712548921759)3)3000+21911761im(1(123i2)125489217593375000000+4605739880729307i22500003)2250000)3.3757694+3.010141iL=46811500+21911761re(1(12+3i2)125489217593375000000+4605739880729307i22500003)22500004681cos(arctan(1500460573988072930712548921759)3)300046813sin(arctan(1500460573988072930712548921759)3)3000+i(21911761im(1(12+3i2)125489217593375000000+4605739880729307i22500003)22500004681sin(arctan(1500460573988072930712548921759)3)3000+46813cos(arctan(1500460573988072930712548921759)3)3000)8.3937517+7.010142i\begin{aligned}L &= - \frac{4681}{1500} + \frac{4681 \cos{\left(\frac{\arctan{\left(\frac{1500 \sqrt{4605739880729307}}{12548921759} \right)}}{3} \right)}}{750} \approx 2.4075211\\L &= - \frac{4681}{1500} - \frac{4681 \cos{\left(\frac{\arctan{\left(\frac{1500 \sqrt{4605739880729307}}{12548921759} \right)}}{3} \right)}}{3000} + \frac{21911761 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{12548921759}{3375000000} + \frac{\sqrt{4605739880729307} i}{2250000}}}\right)}}{2250000} + \frac{4681 \sqrt{3} \sin{\left(\frac{\arctan{\left(\frac{1500 \sqrt{4605739880729307}}{12548921759} \right)}}{3} \right)}}{3000} + i \left(- \frac{4681 \sqrt{3} \cos{\left(\frac{\arctan{\left(\frac{1500 \sqrt{4605739880729307}}{12548921759} \right)}}{3} \right)}}{3000} - \frac{4681 \sin{\left(\frac{\arctan{\left(\frac{1500 \sqrt{4605739880729307}}{12548921759} \right)}}{3} \right)}}{3000} + \frac{21911761 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{12548921759}{3375000000} + \frac{\sqrt{4605739880729307} i}{2250000}}}\right)}}{2250000}\right) \approx -3.3757694 + 3.0 \cdot 10^{-141} i\\L &= - \frac{4681}{1500} + \frac{21911761 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{12548921759}{3375000000} + \frac{\sqrt{4605739880729307} i}{2250000}}}\right)}}{2250000} - \frac{4681 \cos{\left(\frac{\arctan{\left(\frac{1500 \sqrt{4605739880729307}}{12548921759} \right)}}{3} \right)}}{3000} - \frac{4681 \sqrt{3} \sin{\left(\frac{\arctan{\left(\frac{1500 \sqrt{4605739880729307}}{12548921759} \right)}}{3} \right)}}{3000} + i \left(\frac{21911761 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{12548921759}{3375000000} + \frac{\sqrt{4605739880729307} i}{2250000}}}\right)}}{2250000} - \frac{4681 \sin{\left(\frac{\arctan{\left(\frac{1500 \sqrt{4605739880729307}}{12548921759} \right)}}{3} \right)}}{3000} + \frac{4681 \sqrt{3} \cos{\left(\frac{\arctan{\left(\frac{1500 \sqrt{4605739880729307}}{12548921759} \right)}}{3} \right)}}{3000}\right) \approx -8.3937517 + 7.0 \cdot 10^{-142} i\end{aligned}


ii is the imaginary unit, defined as i2=1i^2 = -1.