Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2-8=(x-4)^2
-
Equation 36-x^2=0
-
Equation 4-x=1
-
Equation (2-c)+3(c-3)=-13
- Express {x} in terms of y where:
- 8*x+5*y=-12
- -7*x-4*y=-4
- -2*x-10*y=20
- -1*x+11*y=16
- Factor polynomial:
- x^6-1
-
Identical expressions
- x^ six - one = zero
- x to the power of 6 minus 1 equally 0
- x to the power of six minus one equally zero
- x6-1=0
- x⁶-1=0
- x^6-1=O
-
Similar expressions
- x^6+1=0
x^6-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$x^{6} - 1 = 0$$
Because equation degree is equal to = 6 - contains the even number 6 in the numerator, then
the equation has two real roots.
Get the root 6-th degree of the equation sides:
We get:
$$\sqrt[6]{x^{6}} = \sqrt[6]{1}$$
$$\sqrt[6]{x^{6}} = \left(-1\right) \sqrt[6]{1}$$
or
$$x = 1$$
$$x = -1$$
We get the answer: x = 1
We get the answer: x = -1
or
$$x_{1} = -1$$
$$x_{2} = 1$$
All other 4 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{6} = 1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{6} e^{6 i p} = 1$$
where
$$r = 1$$
- 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
$$p = \frac{\pi N}{3}$$
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:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$z_{3} = - \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$z_{4} = - \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
$$z_{5} = \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$z_{6} = \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{3} = - \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$x_{4} = - \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
$$x_{5} = \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$x_{6} = \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
$$x^{6} - 1 = 0$$
Because equation degree is equal to = 6 - contains the even number 6 in the numerator, then
the equation has two real roots.
Get the root 6-th degree of the equation sides:
We get:
$$\sqrt[6]{x^{6}} = \sqrt[6]{1}$$
$$\sqrt[6]{x^{6}} = \left(-1\right) \sqrt[6]{1}$$
or
$$x = 1$$
$$x = -1$$
We get the answer: x = 1
We get the answer: x = -1
or
$$x_{1} = -1$$
$$x_{2} = 1$$
All other 4 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{6} = 1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{6} e^{6 i p} = 1$$
where
$$r = 1$$
- 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
$$p = \frac{\pi N}{3}$$
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:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$z_{3} = - \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$z_{4} = - \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
$$z_{5} = \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$z_{6} = \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{3} = - \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$x_{4} = - \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
$$x_{5} = \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$x_{6} = \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
Sum and product of roots
[src]
sum
___ ___ ___ ___
1 I*\/ 3 1 I*\/ 3 1 I*\/ 3 1 I*\/ 3
-1 + 1 + - - - ------- + - - + ------- + - - ------- + - + -------
2 2 2 2 2 2 2 2
$$\left(\left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right) + \left(\left(\left(-1 + 1\right) + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)\right) + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
0
$$0$$
product
/ ___\ / ___\ / ___\ / ___\ | 1 I*\/ 3 | | 1 I*\/ 3 | |1 I*\/ 3 | |1 I*\/ 3 | -|- - - -------|*|- - + -------|*|- - -------|*|- + -------| \ 2 2 / \ 2 2 / \2 2 / \2 2 /
$$- (- \frac{1}{2} - \frac{\sqrt{3} i}{2}) \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
-1
$$-1$$
-1
Rapid solution
[src]
x1 = -1
$$x_{1} = -1$$
x2 = 1
$$x_{2} = 1$$
___
1 I*\/ 3
x3 = - - - -------
2 2
$$x_{3} = - \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
___
1 I*\/ 3
x4 = - - + -------
2 2
$$x_{4} = - \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
___
1 I*\/ 3
x5 = - - -------
2 2
$$x_{5} = \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
___
1 I*\/ 3
x6 = - + -------
2 2
$$x_{6} = \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
x6 = 1/2 + sqrt(3)*i/2
Numerical answer
[src]
x1 = -0.5 + 0.866025403784439*i
x2 = -1.0
x3 = 0.5 - 0.866025403784439*i
x4 = 1.0
x5 = -0.5 - 0.866025403784439*i
x6 = 0.5 + 0.866025403784439*i
x6 = 0.5 + 0.866025403784439*i
The graph