Mister Exam
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How to use it?
- Equation:
-
Equation x^6=-3
- Equation 15x^2+7x-2=0
- Equation (2+1.5x)-0.5x=24
- Equation (3*x*x*(x^4-6*x*x+10)-x^3*(4*x^3-12*x))/(x^4-6*x^2+10)^2=0=0
- Express {x} in terms of y where:
- -19*x-4*y=6
- -4*x+3*y=15
- -3*x-18*y=-3
- -5*x+20*y=19
- Graphing y =:
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x^6
- Integral of d{x}:
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x^6
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-3
- Derivative of:
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x^6
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-3
- Sum of series:
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-3
-
Identical expressions
- x^ six =- three
- x to the power of 6 equally minus 3
- x to the power of six equally minus three
- x6=-3
- x⁶=-3
-
Similar expressions
- x^6=+3
x^6=-3 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$x^{6} = -3$$
Because equation degree is equal to = 6 and the free term = -3 < 0,
so the real solutions of the equation d'not exist
All other 6 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{6} = -3$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{6} e^{6 i p} = -3$$
where
$$r = \sqrt[6]{3}$$
- the magnitude of the complex number
Substitute r:
$$e^{6 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = -1$$
so
$$\cos{\left(6 p \right)} = -1$$
and
$$\sin{\left(6 p \right)} = 0$$
then
$$p = \frac{\pi N}{3} + \frac{\pi}{6}$$
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:
$$z_{1} = - \sqrt[6]{3} i$$
$$z_{2} = \sqrt[6]{3} i$$
$$z_{3} = - \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}$$
$$z_{4} = - \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}$$
$$z_{5} = \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}$$
$$z_{6} = \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \sqrt[6]{3} i$$
$$x_{2} = \sqrt[6]{3} i$$
$$x_{3} = - \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}$$
$$x_{4} = - \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}$$
$$x_{5} = \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}$$
$$x_{6} = \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}$$
$$x^{6} = -3$$
Because equation degree is equal to = 6 and the free term = -3 < 0,
so the real solutions of the equation d'not exist
All other 6 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{6} = -3$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{6} e^{6 i p} = -3$$
where
$$r = \sqrt[6]{3}$$
- the magnitude of the complex number
Substitute r:
$$e^{6 i p} = -1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = -1$$
so
$$\cos{\left(6 p \right)} = -1$$
and
$$\sin{\left(6 p \right)} = 0$$
then
$$p = \frac{\pi N}{3} + \frac{\pi}{6}$$
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:
$$z_{1} = - \sqrt[6]{3} i$$
$$z_{2} = \sqrt[6]{3} i$$
$$z_{3} = - \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}$$
$$z_{4} = - \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}$$
$$z_{5} = \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}$$
$$z_{6} = \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \sqrt[6]{3} i$$
$$x_{2} = \sqrt[6]{3} i$$
$$x_{3} = - \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}$$
$$x_{4} = - \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}$$
$$x_{5} = \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}$$
$$x_{6} = \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}$$
Rapid solution
[src]
6 ___ x1 = -I*\/ 3
$$x_{1} = - \sqrt[6]{3} i$$
6 ___ x2 = I*\/ 3
$$x_{2} = \sqrt[6]{3} i$$
2/3 6 ___
3 I*\/ 3
x3 = - ---- - -------
2 2
$$x_{3} = - \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}$$
2/3 6 ___
3 I*\/ 3
x4 = - ---- + -------
2 2
$$x_{4} = - \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}$$
2/3 6 ___
3 I*\/ 3
x5 = ---- - -------
2 2
$$x_{5} = \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}$$
2/3 6 ___
3 I*\/ 3
x6 = ---- + -------
2 2
$$x_{6} = \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}$$
x6 = 3^(2/3)/2 + 3^(1/6)*i/2
Sum and product of roots
[src]
sum
2/3 6 ___ 2/3 6 ___ 2/3 6 ___ 2/3 6 ___
6 ___ 6 ___ 3 I*\/ 3 3 I*\/ 3 3 I*\/ 3 3 I*\/ 3
- I*\/ 3 + I*\/ 3 + - ---- - ------- + - ---- + ------- + ---- - ------- + ---- + -------
2 2 2 2 2 2 2 2
$$\left(\left(\frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}\right) + \left(\left(\left(- \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}\right) + \left(- \sqrt[6]{3} i + \sqrt[6]{3} i\right)\right) + \left(- \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}\right)\right)\right) + \left(\frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}\right)$$
=
0
$$0$$
product
/ 2/3 6 ___\ / 2/3 6 ___\ / 2/3 6 ___\ / 2/3 6 ___\
6 ___ 6 ___ | 3 I*\/ 3 | | 3 I*\/ 3 | |3 I*\/ 3 | |3 I*\/ 3 |
-I*\/ 3 *I*\/ 3 *|- ---- - -------|*|- ---- + -------|*|---- - -------|*|---- + -------|
\ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$- \sqrt[6]{3} i \sqrt[6]{3} i \left(- \frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}\right) \left(- \frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}\right) \left(\frac{3^{\frac{2}{3}}}{2} - \frac{\sqrt[6]{3} i}{2}\right) \left(\frac{3^{\frac{2}{3}}}{2} + \frac{\sqrt[6]{3} i}{2}\right)$$
=
3
$$3$$
3
Numerical answer
[src]
x1 = -1.04004191152595 + 0.600468477588001*i
x2 = 1.04004191152595 + 0.600468477588001*i
x3 = 1.04004191152595 - 0.600468477588001*i
x4 = -1.200936955176*i
x5 = 1.200936955176*i
x6 = -1.04004191152595 - 0.600468477588001*i
x6 = -1.04004191152595 - 0.600468477588001*i
The graph