Mister Exam

Other calculators

x^-5x+3=0 equation

The teacher will be very surprised to see your correct solution 😉

v

Numerical solution:

Do search numerical solution at [, ]

The solution

You have entered [src]
x         
-- + 3 = 0
 5        
x         
$$\frac{x}{x^{5}} + 3 = 0$$
Detail solution
Given the equation
$$\frac{x}{x^{5}} + 3 = 0$$
Because equation degree is equal to = -4 and the free term = -3 < 0,
so the real solutions of the equation d'not exist

All other 4 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$\frac{1}{z^{4}} = -3$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$\frac{e^{- 4 i p}}{r^{4}} = -3$$
where
$$r = \frac{3^{\frac{3}{4}}}{3}$$
- the magnitude of the complex number
Substitute r:
$$e^{- 4 i p} = -1$$
Using Euler’s formula, we find roots for p
$$- i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = -1$$
so
$$\cos{\left(4 p \right)} = -1$$
and
$$- \sin{\left(4 p \right)} = 0$$
then
$$p = - \frac{\pi N}{2} - \frac{\pi}{4}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
$$z_{2} = - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
$$z_{3} = \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
$$z_{4} = \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
do backward replacement
$$z = x$$
$$x = z$$

The final answer:
$$x_{1} = - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
$$x_{2} = - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
$$x_{3} = \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
$$x_{4} = \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
The graph
Rapid solution [src]
         ___  3/4       ___  3/4
       \/ 2 *3      I*\/ 2 *3   
x1 = - ---------- - ------------
           6             6      
$$x_{1} = - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
         ___  3/4       ___  3/4
       \/ 2 *3      I*\/ 2 *3   
x2 = - ---------- + ------------
           6             6      
$$x_{2} = - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
       ___  3/4       ___  3/4
     \/ 2 *3      I*\/ 2 *3   
x3 = ---------- - ------------
         6             6      
$$x_{3} = \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
       ___  3/4       ___  3/4
     \/ 2 *3      I*\/ 2 *3   
x4 = ---------- + ------------
         6             6      
$$x_{4} = \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}$$
x4 = sqrt(2)*3^(3/4)/6 + sqrt(2)*3^(3/4)*i/6
Sum and product of roots [src]
sum
    ___  3/4       ___  3/4       ___  3/4       ___  3/4     ___  3/4       ___  3/4     ___  3/4       ___  3/4
  \/ 2 *3      I*\/ 2 *3        \/ 2 *3      I*\/ 2 *3      \/ 2 *3      I*\/ 2 *3      \/ 2 *3      I*\/ 2 *3   
- ---------- - ------------ + - ---------- + ------------ + ---------- - ------------ + ---------- + ------------
      6             6               6             6             6             6             6             6      
$$\left(\left(\frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}\right) + \left(\left(- \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}\right) + \left(- \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}\right)\right)\right) + \left(\frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}\right)$$
=
0
$$0$$
product
/    ___  3/4       ___  3/4\ /    ___  3/4       ___  3/4\ /  ___  3/4       ___  3/4\ /  ___  3/4       ___  3/4\
|  \/ 2 *3      I*\/ 2 *3   | |  \/ 2 *3      I*\/ 2 *3   | |\/ 2 *3      I*\/ 2 *3   | |\/ 2 *3      I*\/ 2 *3   |
|- ---------- - ------------|*|- ---------- + ------------|*|---------- - ------------|*|---------- + ------------|
\      6             6      / \      6             6      / \    6             6      / \    6             6      /
$$\left(- \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}\right) \left(- \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}\right) \left(\frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}\right) \left(\frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{6}\right)$$
=
1/3
$$\frac{1}{3}$$
1/3
Numerical answer [src]
x1 = -0.537284965911771 - 0.537284965911771*i
x2 = -0.537284965911771 + 0.537284965911771*i
x3 = 0.537284965911771 - 0.537284965911771*i
x4 = 0.537284965911771 + 0.537284965911771*i
x4 = 0.537284965911771 + 0.537284965911771*i