(L^3)+9.362(L^2)-68.218=0
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You asked:
Solve the equation
L 3 + 9.362 L 2 − 68.218 = 0 {L}^{3} + 9.362 {L}^{2} - 68.218 = 0 L 3 + 9.362 L 2 − 68.218 = 0 for the variable
L L L .
MathBot Answer:
The 3 solutions to the equation are:
L = − 4681 1500 + 4681 cos ( arctan ( 1500 4605739880729307 12548921759 ) 3 ) 750 ≈ 2.4075211 L = − 4681 1500 − 4681 cos ( arctan ( 1500 4605739880729307 12548921759 ) 3 ) 3000 + 21911761 re ( 1 ( − 1 2 − 3 i 2 ) 12548921759 3375000000 + 4605739880729307 i 2250000 3 ) 2250000 + 4681 3 sin ( arctan ( 1500 4605739880729307 12548921759 ) 3 ) 3000 + i ( − 4681 3 cos ( arctan ( 1500 4605739880729307 12548921759 ) 3 ) 3000 − 4681 sin ( arctan ( 1500 4605739880729307 12548921759 ) 3 ) 3000 + 21911761 im ( 1 ( − 1 2 − 3 i 2 ) 12548921759 3375000000 + 4605739880729307 i 2250000 3 ) 2250000 ) ≈ − 3.3757694 + 3.0 ⋅ 1 0 − 141 i L = − 4681 1500 + 21911761 re ( 1 ( − 1 2 + 3 i 2 ) 12548921759 3375000000 + 4605739880729307 i 2250000 3 ) 2250000 − 4681 cos ( arctan ( 1500 4605739880729307 12548921759 ) 3 ) 3000 − 4681 3 sin ( arctan ( 1500 4605739880729307 12548921759 ) 3 ) 3000 + i ( 21911761 im ( 1 ( − 1 2 + 3 i 2 ) 12548921759 3375000000 + 4605739880729307 i 2250000 3 ) 2250000 − 4681 sin ( arctan ( 1500 4605739880729307 12548921759 ) 3 ) 3000 + 4681 3 cos ( arctan ( 1500 4605739880729307 12548921759 ) 3 ) 3000 ) ≈ − 8.3937517 + 7.0 ⋅ 1 0 − 142 i \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} L L L = − 1500 4681 + 750 4681 cos ( 3 a r c t a n ( 12548921759 1500 4605739880729307 ) ) ≈ 2.4075211 = − 1500 4681 − 3000 4681 cos ( 3 a r c t a n ( 12548921759 1500 4605739880729307 ) ) + 2250000 21911761 re ( ( − 2 1 − 2 3 i ) 3 3375000000 12548921759 + 2250000 4605739880729307 i 1 ) + 3000 4681 3 sin ( 3 a r c t a n ( 12548921759 1500 4605739880729307 ) ) + i − 3000 4681 3 cos ( 3 a r c t a n ( 12548921759 1500 4605739880729307 ) ) − 3000 4681 sin ( 3 a r c t a n ( 12548921759 1500 4605739880729307 ) ) + 2250000 21911761 im ( ( − 2 1 − 2 3 i ) 3 3375000000 12548921759 + 2250000 4605739880729307 i 1 ) ≈ − 3.3757694 + 3.0 ⋅ 1 0 − 141 i = − 1500 4681 + 2250000 21911761 re ( ( − 2 1 + 2 3 i ) 3 3375000000 12548921759 + 2250000 4605739880729307 i 1 ) − 3000 4681 cos ( 3 a r c t a n ( 12548921759 1500 4605739880729307 ) ) − 3000 4681 3 sin ( 3 a r c t a n ( 12548921759 1500 4605739880729307 ) ) + i 2250000 21911761 im ( ( − 2 1 + 2 3 i ) 3 3375000000 12548921759 + 2250000 4605739880729307 i 1 ) − 3000 4681 sin ( 3 a r c t a n ( 12548921759 1500 4605739880729307 ) ) + 3000 4681 3 cos ( 3 a r c t a n ( 12548921759 1500 4605739880729307 ) ) ≈ − 8.3937517 + 7.0 ⋅ 1 0 − 142 i
i i i imaginary unit , defined as i 2 = − 1 i^2 = -1 i 2 = − 1
