1.5x −2(x−50.0)^3−74.5 = 0.75
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You asked:
Solve the equation
1.5 x − 2 ( x − 50.0 ) 3 − 74.5 = 0.75 1.5 x - 2 {\left( x - 50.0 \right)}^{3} - 74.5 = 0.75 1.5 x − 2 ( x − 50.0 ) 3 − 74.5 = 0.75 for the variable
x x x .
MathBot Answer:
The 3 solutions to the equation are:
x = 50 − cos ( π 9 ) ≈ 49.060307 x = − 3 sin ( π 9 ) 4 − 3 re ( 1 ( − 1 2 − 3 i 2 ) 27 16 + 27 3 i 16 3 ) 4 + cos ( π 9 ) 4 + 50 + i ( − 3 im ( 1 ( − 1 2 − 3 i 2 ) 27 16 + 27 3 i 16 3 ) 4 + sin ( π 9 ) 4 + 3 cos ( π 9 ) 4 ) ≈ 50.173648 − 2.0 ⋅ 1 0 − 143 i x = 3 sin ( π 9 ) 4 + cos ( π 9 ) 4 − 3 re ( 1 ( − 1 2 + 3 i 2 ) 27 16 + 27 3 i 16 3 ) 4 + 50 + i ( − 3 cos ( π 9 ) 4 + sin ( π 9 ) 4 − 3 im ( 1 ( − 1 2 + 3 i 2 ) 27 16 + 27 3 i 16 3 ) 4 ) ≈ 50.766044 + 2.0 ⋅ 1 0 − 142 i \begin{aligned}x &= 50 - \cos{\left(\frac{\pi}{9} \right)} \approx 49.060307\\x &= - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{4} - \frac{3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{16} + \frac{27 \sqrt{3} i}{16}}}\right)}}{4} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{4} + 50 + i \left(- \frac{3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{16} + \frac{27 \sqrt{3} i}{16}}}\right)}}{4} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{4} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{4}\right) \approx 50.173648 -2.0 \cdot 10^{-143} i\\x &= \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{4} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{4} - \frac{3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{16} + \frac{27 \sqrt{3} i}{16}}}\right)}}{4} + 50 + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{4} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{4} - \frac{3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{16} + \frac{27 \sqrt{3} i}{16}}}\right)}}{4}\right) \approx 50.766044 + 2.0 \cdot 10^{-142} i\end{aligned} x x x = 50 − cos ( 9 π ) ≈ 49.060307 = − 4 3 sin ( 9 π ) − 4 3 re ( ( − 2 1 − 2 3 i ) 3 16 27 + 16 27 3 i 1 ) + 4 cos ( 9 π ) + 50 + i − 4 3 im ( ( − 2 1 − 2 3 i ) 3 16 27 + 16 27 3 i 1 ) + 4 sin ( 9 π ) + 4 3 cos ( 9 π ) ≈ 50.173648 − 2.0 ⋅ 1 0 − 143 i = 4 3 sin ( 9 π ) + 4 cos ( 9 π ) − 4 3 re ( ( − 2 1 + 2 3 i ) 3 16 27 + 16 27 3 i 1 ) + 50 + i − 4 3 cos ( 9 π ) + 4 sin ( 9 π ) − 4 3 im ( ( − 2 1 + 2 3 i ) 3 16 27 + 16 27 3 i 1 ) ≈ 50.766044 + 2.0 ⋅ 1 0 − 142 i
i i i imaginary unit , defined as i 2 = − 1 i^2 = -1 i 2 = − 1
