2*x^3-46*x^2-1024*x+9486=0
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You asked:
Solve the equation
2 x 3 − 46 x 2 − 1024 x + 9486 = 0 2 {x}^{3} - 46 {x}^{2} - 1024 x + 9486 = 0 2 x 3 − 46 x 2 − 1024 x + 9486 = 0 for the variable
x x x .
MathBot Answer:
The 3 solutions to the equation are:
x = 23 3 + 2 2065 cos ( arctan ( 9 434782771 2257 ) 3 ) 3 ≈ 33.963286 x = − 2065 cos ( arctan ( 9 434782771 2257 ) 3 ) 6 + 2065 re ( 1 ( − 1 2 − 3 i 2 ) 2257 54 + 434782771 i 6 3 ) 9 + 6195 sin ( arctan ( 9 434782771 2257 ) 3 ) 6 + 23 3 + i ( − 6195 cos ( arctan ( 9 434782771 2257 ) 3 ) 6 − 2065 sin ( arctan ( 9 434782771 2257 ) 3 ) 6 + 2065 im ( 1 ( − 1 2 − 3 i 2 ) 2257 54 + 434782771 i 6 3 ) 9 ) ≈ 7.545222 + 5.0 ⋅ 1 0 − 141 i x = 2065 re ( 1 ( − 1 2 + 3 i 2 ) 2257 54 + 434782771 i 6 3 ) 9 − 2065 cos ( arctan ( 9 434782771 2257 ) 3 ) 6 − 6195 sin ( arctan ( 9 434782771 2257 ) 3 ) 6 + 23 3 + i ( 2065 im ( 1 ( − 1 2 + 3 i 2 ) 2257 54 + 434782771 i 6 3 ) 9 − 2065 sin ( arctan ( 9 434782771 2257 ) 3 ) 6 + 6195 cos ( arctan ( 9 434782771 2257 ) 3 ) 6 ) ≈ − 18.508508 − 1.0 ⋅ 1 0 − 141 i \begin{aligned}x &= \frac{23}{3} + \frac{2 \sqrt{2065} \cos{\left(\frac{\arctan{\left(\frac{9 \sqrt{434782771}}{2257} \right)}}{3} \right)}}{3} \approx 33.963286\\x &= - \frac{\sqrt{2065} \cos{\left(\frac{\arctan{\left(\frac{9 \sqrt{434782771}}{2257} \right)}}{3} \right)}}{6} + \frac{2065 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{2257}{54} + \frac{\sqrt{434782771} i}{6}}}\right)}}{9} + \frac{\sqrt{6195} \sin{\left(\frac{\arctan{\left(\frac{9 \sqrt{434782771}}{2257} \right)}}{3} \right)}}{6} + \frac{23}{3} + i \left(- \frac{\sqrt{6195} \cos{\left(\frac{\arctan{\left(\frac{9 \sqrt{434782771}}{2257} \right)}}{3} \right)}}{6} - \frac{\sqrt{2065} \sin{\left(\frac{\arctan{\left(\frac{9 \sqrt{434782771}}{2257} \right)}}{3} \right)}}{6} + \frac{2065 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{2257}{54} + \frac{\sqrt{434782771} i}{6}}}\right)}}{9}\right) \approx 7.545222 + 5.0 \cdot 10^{-141} i\\x &= \frac{2065 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{2257}{54} + \frac{\sqrt{434782771} i}{6}}}\right)}}{9} - \frac{\sqrt{2065} \cos{\left(\frac{\arctan{\left(\frac{9 \sqrt{434782771}}{2257} \right)}}{3} \right)}}{6} - \frac{\sqrt{6195} \sin{\left(\frac{\arctan{\left(\frac{9 \sqrt{434782771}}{2257} \right)}}{3} \right)}}{6} + \frac{23}{3} + i \left(\frac{2065 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{2257}{54} + \frac{\sqrt{434782771} i}{6}}}\right)}}{9} - \frac{\sqrt{2065} \sin{\left(\frac{\arctan{\left(\frac{9 \sqrt{434782771}}{2257} \right)}}{3} \right)}}{6} + \frac{\sqrt{6195} \cos{\left(\frac{\arctan{\left(\frac{9 \sqrt{434782771}}{2257} \right)}}{3} \right)}}{6}\right) \approx -18.508508 -1.0 \cdot 10^{-141} i\end{aligned} x x x = 3 23 + 3 2 2065 cos ( 3 a r c t a n ( 2257 9 434782771 ) ) ≈ 33.963286 = − 6 2065 cos ( 3 a r c t a n ( 2257 9 434782771 ) ) + 9 2065 re ( ( − 2 1 − 2 3 i ) 3 54 2257 + 6 434782771 i 1 ) + 6 6195 sin ( 3 a r c t a n ( 2257 9 434782771 ) ) + 3 23 + i − 6 6195 cos ( 3 a r c t a n ( 2257 9 434782771 ) ) − 6 2065 sin ( 3 a r c t a n ( 2257 9 434782771 ) ) + 9 2065 im ( ( − 2 1 − 2 3 i ) 3 54 2257 + 6 434782771 i 1 ) ≈ 7.545222 + 5.0 ⋅ 1 0 − 141 i = 9 2065 re ( ( − 2 1 + 2 3 i ) 3 54 2257 + 6 434782771 i 1 ) − 6 2065 cos ( 3 a r c t a n ( 2257 9 434782771 ) ) − 6 6195 sin ( 3 a r c t a n ( 2257 9 434782771 ) ) + 3 23 + i 9 2065 im ( ( − 2 1 + 2 3 i ) 3 54 2257 + 6 434782771 i 1 ) − 6 2065 sin ( 3 a r c t a n ( 2257 9 434782771 ) ) + 6 6195 cos ( 3 a r c t a n ( 2257 9 434782771 ) ) ≈ − 18.508508 − 1.0 ⋅ 1 0 − 141 i
i i i imaginary unit , defined as i 2 = − 1 i^2 = -1 i 2 = − 1
