Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation -x/12+x=55/12
-
Equation x^2-2x+5=0
-
Equation cos(x)=2/3
-
Equation 4x+1=-3x+15
- Express {x} in terms of y where:
- -4*x+6*y=-7
- 2*x+3*y=14
- 18*x-8*y=-2
- 10*x-16*y=-6
- Derivative of:
-
x^4+1
- Graphing y =:
- x^4+1
- Factor polynomial:
- x^4+1
-
Identical expressions
- x^ four + one = zero
- x to the power of 4 plus 1 equally 0
- x to the power of four plus one equally zero
- x4+1=0
- x⁴+1=0
- x^4+1=O
-
Similar expressions
- x^4-1=0
x^4+1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$x^{4} + 1 = 0$$
Because equation degree is equal to = 4 and the free term = -1 < 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:
$$z^{4} = -1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{4} e^{4 i p} = -1$$
where
$$r = 1$$
- 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:
$$z_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$z_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$x_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$x_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$x_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$x^{4} + 1 = 0$$
Because equation degree is equal to = 4 and the free term = -1 < 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:
$$z^{4} = -1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{4} e^{4 i p} = -1$$
where
$$r = 1$$
- 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:
$$z_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$z_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$z_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$x_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
$$x_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
$$x_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
Sum and product of roots
[src]
sum
___ ___ ___ ___ ___ ___ ___ ___
\/ 2 I*\/ 2 \/ 2 I*\/ 2 \/ 2 I*\/ 2 \/ 2 I*\/ 2
- ----- - ------- + - ----- + ------- + ----- - ------- + ----- + -------
2 2 2 2 2 2 2 2
$$\left(\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) + \left(\left(- \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) + \left(- \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right)\right)\right) + \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right)$$
=
0
$$0$$
product
/ ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ | \/ 2 I*\/ 2 | | \/ 2 I*\/ 2 | |\/ 2 I*\/ 2 | |\/ 2 I*\/ 2 | |- ----- - -------|*|- ----- + -------|*|----- - -------|*|----- + -------| \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$\left(- \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) \left(- \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right) \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right)$$
=
1
$$1$$
1
Rapid solution
[src]
___ ___
\/ 2 I*\/ 2
x1 = - ----- - -------
2 2
$$x_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
___ ___
\/ 2 I*\/ 2
x2 = - ----- + -------
2 2
$$x_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
___ ___
\/ 2 I*\/ 2
x3 = ----- - -------
2 2
$$x_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
___ ___
\/ 2 I*\/ 2
x4 = ----- + -------
2 2
$$x_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
x4 = sqrt(2)/2 + sqrt(2)*i/2
Numerical answer
[src]
x1 = -0.707106781186548 - 0.707106781186548*i
x2 = 0.707106781186548 + 0.707106781186548*i
x3 = -0.707106781186548 + 0.707106781186548*i
x4 = 0.707106781186548 - 0.707106781186548*i
x4 = 0.707106781186548 - 0.707106781186548*i
The graph