7 over 3-x equals 5 over x+3 + 4 over x to the power 2 - 9
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You asked:
Solve the equation
7 3 − x = 5 x + 3 + 4 x 2 − 9 \frac{7}{3} - x = \frac{5}{x} + 3 + \frac{4}{{x}^{2}} - 9 3 7 − x = x 5 + 3 + x 2 4 − 9 for the variable
x x x .
MathBot Answer:
The 3 solutions to the equation are:
x = 25 9 + 14 10 cos ( arctan ( 27 190711 18209 ) 3 ) 9 ≈ 7.6069099 x = − 7 10 cos ( arctan ( 27 190711 18209 ) 3 ) 18 + 490 re ( 1 ( − 1 2 − 3 i 2 ) 18209 1458 + 190711 i 54 3 ) 81 + 7 30 sin ( arctan ( 27 190711 18209 ) 3 ) 18 + 25 9 + i ( − 7 30 cos ( arctan ( 27 190711 18209 ) 3 ) 18 − 7 10 sin ( arctan ( 27 190711 18209 ) 3 ) 18 + 490 im ( 1 ( − 1 2 − 3 i 2 ) 18209 1458 + 190711 i 54 3 ) 81 ) ≈ 1.174236 + 2.0 ⋅ 1 0 − 142 i x = 490 re ( 1 ( − 1 2 + 3 i 2 ) 18209 1458 + 190711 i 54 3 ) 81 − 7 10 cos ( arctan ( 27 190711 18209 ) 3 ) 18 − 7 30 sin ( arctan ( 27 190711 18209 ) 3 ) 18 + 25 9 + i ( 490 im ( 1 ( − 1 2 + 3 i 2 ) 18209 1458 + 190711 i 54 3 ) 81 − 7 10 sin ( arctan ( 27 190711 18209 ) 3 ) 18 + 7 30 cos ( arctan ( 27 190711 18209 ) 3 ) 18 ) ≈ − 0.4478126 + 2.0 ⋅ 1 0 − 143 i \begin{aligned}x &= \frac{25}{9} + \frac{14 \sqrt{10} \cos{\left(\frac{\arctan{\left(\frac{27 \sqrt{190711}}{18209} \right)}}{3} \right)}}{9} \approx 7.6069099\\x &= - \frac{7 \sqrt{10} \cos{\left(\frac{\arctan{\left(\frac{27 \sqrt{190711}}{18209} \right)}}{3} \right)}}{18} + \frac{490 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{18209}{1458} + \frac{\sqrt{190711} i}{54}}}\right)}}{81} + \frac{7 \sqrt{30} \sin{\left(\frac{\arctan{\left(\frac{27 \sqrt{190711}}{18209} \right)}}{3} \right)}}{18} + \frac{25}{9} + i \left(- \frac{7 \sqrt{30} \cos{\left(\frac{\arctan{\left(\frac{27 \sqrt{190711}}{18209} \right)}}{3} \right)}}{18} - \frac{7 \sqrt{10} \sin{\left(\frac{\arctan{\left(\frac{27 \sqrt{190711}}{18209} \right)}}{3} \right)}}{18} + \frac{490 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{18209}{1458} + \frac{\sqrt{190711} i}{54}}}\right)}}{81}\right) \approx 1.174236 + 2.0 \cdot 10^{-142} i\\x &= \frac{490 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{18209}{1458} + \frac{\sqrt{190711} i}{54}}}\right)}}{81} - \frac{7 \sqrt{10} \cos{\left(\frac{\arctan{\left(\frac{27 \sqrt{190711}}{18209} \right)}}{3} \right)}}{18} - \frac{7 \sqrt{30} \sin{\left(\frac{\arctan{\left(\frac{27 \sqrt{190711}}{18209} \right)}}{3} \right)}}{18} + \frac{25}{9} + i \left(\frac{490 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{18209}{1458} + \frac{\sqrt{190711} i}{54}}}\right)}}{81} - \frac{7 \sqrt{10} \sin{\left(\frac{\arctan{\left(\frac{27 \sqrt{190711}}{18209} \right)}}{3} \right)}}{18} + \frac{7 \sqrt{30} \cos{\left(\frac{\arctan{\left(\frac{27 \sqrt{190711}}{18209} \right)}}{3} \right)}}{18}\right) \approx -0.4478126 + 2.0 \cdot 10^{-143} i\end{aligned} x x x = 9 25 + 9 14 10 cos ( 3 a r c t a n ( 18209 27 190711 ) ) ≈ 7.6069099 = − 18 7 10 cos ( 3 a r c t a n ( 18209 27 190711 ) ) + 81 490 re ( ( − 2 1 − 2 3 i ) 3 1458 18209 + 54 190711 i 1 ) + 18 7 30 sin ( 3 a r c t a n ( 18209 27 190711 ) ) + 9 25 + i − 18 7 30 cos ( 3 a r c t a n ( 18209 27 190711 ) ) − 18 7 10 sin ( 3 a r c t a n ( 18209 27 190711 ) ) + 81 490 im ( ( − 2 1 − 2 3 i ) 3 1458 18209 + 54 190711 i 1 ) ≈ 1.174236 + 2.0 ⋅ 1 0 − 142 i = 81 490 re ( ( − 2 1 + 2 3 i ) 3 1458 18209 + 54 190711 i 1 ) − 18 7 10 cos ( 3 a r c t a n ( 18209 27 190711 ) ) − 18 7 30 sin ( 3 a r c t a n ( 18209 27 190711 ) ) + 9 25 + i 81 490 im ( ( − 2 1 + 2 3 i ) 3 1458 18209 + 54 190711 i 1 ) − 18 7 10 sin ( 3 a r c t a n ( 18209 27 190711 ) ) + 18 7 30 cos ( 3 a r c t a n ( 18209 27 190711 ) ) ≈ − 0.4478126 + 2.0 ⋅ 1 0 − 143 i
i i i imaginary unit , defined as i 2 = − 1 i^2 = -1 i 2 = − 1
