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x^4+10*x+9=0 equation

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Numerical solution:

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The solution

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 4               
x  + 10*x + 9 = 0
(x4+10x)+9=0\left(x^{4} + 10 x\right) + 9 = 0
Simplify
The graph
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.5-2000020000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = -1
x1=−1x_{1} = -1 
                                     _________________     /                                  _________________\
                                  3 /           _____      |          ___              ___ 3 /           _____ |
     1             1              \/  125 + 9*\/ 193       |        \/ 3             \/ 3 *\/  125 + 9*\/ 193  |
x2 = - - ---------------------- + -------------------- + I*|---------------------- + --------------------------|
     3        _________________            6               |     _________________               6             |
           3 /           _____                             |  3 /           _____                              |
         3*\/  125 + 9*\/ 193                              \3*\/  125 + 9*\/ 193                               /
x2=−13125+91933+13+125+919336+i(33125+91933+3125+919336)x_{2} = - \frac{1}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{1}{3} + \frac{\sqrt[3]{125 + 9 \sqrt{193}}}{6} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{\sqrt{3} \sqrt[3]{125 + 9 \sqrt{193}}}{6}\right) 
                                     _________________     /                                    _________________\
                                  3 /           _____      |            ___              ___ 3 /           _____ |
     1             1              \/  125 + 9*\/ 193       |          \/ 3             \/ 3 *\/  125 + 9*\/ 193  |
x3 = - - ---------------------- + -------------------- + I*|- ---------------------- - --------------------------|
     3        _________________            6               |       _________________               6             |
           3 /           _____                             |    3 /           _____                              |
         3*\/  125 + 9*\/ 193                              \  3*\/  125 + 9*\/ 193                               /
x3=−13125+91933+13+125+919336+i(−3125+919336−33125+91933)x_{3} = - \frac{1}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{1}{3} + \frac{\sqrt[3]{125 + 9 \sqrt{193}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{125 + 9 \sqrt{193}}}{6} - \frac{\sqrt{3}}{3 \sqrt[3]{125 + 9 \sqrt{193}}}\right) 
            _________________                         
         3 /           _____                          
     1   \/  125 + 9*\/ 193               2           
x4 = - - -------------------- + ----------------------
     3            3                  _________________
                                  3 /           _____ 
                                3*\/  125 + 9*\/ 193
x4=−125+919333+23125+91933+13x_{4} = - \frac{\sqrt[3]{125 + 9 \sqrt{193}}}{3} + \frac{2}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{1}{3} 
x4 = -(125 + 9*sqrt(193))^(1/3)/3 + 2/(3*(125 + 9*sqrt(193))^(1/3)) + 1/3
Sum and product of roots [src] 
sum
                                     _________________     /                                  _________________\                                   _________________     /                                    _________________\          _________________                         
                                  3 /           _____      |          ___              ___ 3 /           _____ |                                3 /           _____      |            ___              ___ 3 /           _____ |       3 /           _____                          
     1             1              \/  125 + 9*\/ 193       |        \/ 3             \/ 3 *\/  125 + 9*\/ 193  |   1             1              \/  125 + 9*\/ 193       |          \/ 3             \/ 3 *\/  125 + 9*\/ 193  |   1   \/  125 + 9*\/ 193               2           
-1 + - - ---------------------- + -------------------- + I*|---------------------- + --------------------------| + - - ---------------------- + -------------------- + I*|- ---------------------- - --------------------------| + - - -------------------- + ----------------------
     3        _________________            6               |     _________________               6             |   3        _________________            6               |       _________________               6             |   3            3                  _________________
           3 /           _____                             |  3 /           _____                              |         3 /           _____                             |    3 /           _____                              |                                3 /           _____ 
         3*\/  125 + 9*\/ 193                              \3*\/  125 + 9*\/ 193                               /       3*\/  125 + 9*\/ 193                              \  3*\/  125 + 9*\/ 193                               /                              3*\/  125 + 9*\/ 193
(−125+919333+23125+91933+13)+((−13125+91933+13+125+919336+i(−3125+919336−33125+91933))+(−1+(−13125+91933+13+125+919336+i(33125+91933+3125+919336))))\left(- \frac{\sqrt[3]{125 + 9 \sqrt{193}}}{3} + \frac{2}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{1}{3}\right) + \left(\left(- \frac{1}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{1}{3} + \frac{\sqrt[3]{125 + 9 \sqrt{193}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{125 + 9 \sqrt{193}}}{6} - \frac{\sqrt{3}}{3 \sqrt[3]{125 + 9 \sqrt{193}}}\right)\right) + \left(-1 + \left(- \frac{1}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{1}{3} + \frac{\sqrt[3]{125 + 9 \sqrt{193}}}{6} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{\sqrt{3} \sqrt[3]{125 + 9 \sqrt{193}}}{6}\right)\right)\right)\right) 
=
  /                                    _________________\     /                                  _________________\
  |            ___              ___ 3 /           _____ |     |          ___              ___ 3 /           _____ |
  |          \/ 3             \/ 3 *\/  125 + 9*\/ 193  |     |        \/ 3             \/ 3 *\/  125 + 9*\/ 193  |
I*|- ---------------------- - --------------------------| + I*|---------------------- + --------------------------|
  |       _________________               6             |     |     _________________               6             |
  |    3 /           _____                              |     |  3 /           _____                              |
  \  3*\/  125 + 9*\/ 193                               /     \3*\/  125 + 9*\/ 193                               /
i(−3125+919336−33125+91933)+i(33125+91933+3125+919336)i \left(- \frac{\sqrt{3} \sqrt[3]{125 + 9 \sqrt{193}}}{6} - \frac{\sqrt{3}}{3 \sqrt[3]{125 + 9 \sqrt{193}}}\right) + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{\sqrt{3} \sqrt[3]{125 + 9 \sqrt{193}}}{6}\right) 
product
 /                                _________________     /                                  _________________\\ /                                _________________     /                                    _________________\\ /       _________________                         \
 |                             3 /           _____      |          ___              ___ 3 /           _____ || |                             3 /           _____      |            ___              ___ 3 /           _____ || |    3 /           _____                          |
 |1             1              \/  125 + 9*\/ 193       |        \/ 3             \/ 3 *\/  125 + 9*\/ 193  || |1             1              \/  125 + 9*\/ 193       |          \/ 3             \/ 3 *\/  125 + 9*\/ 193  || |1   \/  125 + 9*\/ 193               2           |
-|- - ---------------------- + -------------------- + I*|---------------------- + --------------------------||*|- - ---------------------- + -------------------- + I*|- ---------------------- - --------------------------||*|- - -------------------- + ----------------------|
 |3        _________________            6               |     _________________               6             || |3        _________________            6               |       _________________               6             || |3            3                  _________________|
 |      3 /           _____                             |  3 /           _____                              || |      3 /           _____                             |    3 /           _____                              || |                             3 /           _____ |
 \    3*\/  125 + 9*\/ 193                              \3*\/  125 + 9*\/ 193                               // \    3*\/  125 + 9*\/ 193                              \  3*\/  125 + 9*\/ 193                               // \                           3*\/  125 + 9*\/ 193  /
−(−13125+91933+13+125+919336+i(33125+91933+3125+919336))(−13125+91933+13+125+919336+i(−3125+919336−33125+91933))(−125+919333+23125+91933+13)- (- \frac{1}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{1}{3} + \frac{\sqrt[3]{125 + 9 \sqrt{193}}}{6} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{\sqrt{3} \sqrt[3]{125 + 9 \sqrt{193}}}{6}\right)) \left(- \frac{1}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{1}{3} + \frac{\sqrt[3]{125 + 9 \sqrt{193}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{125 + 9 \sqrt{193}}}{6} - \frac{\sqrt{3}}{3 \sqrt[3]{125 + 9 \sqrt{193}}}\right)\right) \left(- \frac{\sqrt[3]{125 + 9 \sqrt{193}}}{3} + \frac{2}{3 \sqrt[3]{125 + 9 \sqrt{193}}} + \frac{1}{3}\right) 
=
9
99 
9
Numerical answer [src] 
x1 = 1.33040121988528 + 1.9102617040202*i
x2 = 1.33040121988528 - 1.9102617040202*i
x3 = -1.66080243977055
x4 = -1.0
x4 = -1.0
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