Mister Exam
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How to use it?
- Equation:
- Equation x^4+81=0
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Equation 17x+2x^2+21=0
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Equation z^4=1
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- Factor polynomial:
- x^4+81
-
Identical expressions
- x^ four + eighty-one = zero
- x to the power of 4 plus 81 equally 0
- x to the power of four plus eighty minus one equally zero
- x4+81=0
- x⁴+81=0
- x^4+81=O
-
Similar expressions
- x^4-81=0
x^4+81=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$x^{4} + 81 = 0$$
Because equation degree is equal to = 4 and the free term = -81 < 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:
$$z^{4} = -81$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{4} e^{4 i p} = -81$$
where
$$r = 3$$
- 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:
$$z_{1} = - \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$z_{2} = - \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
$$z_{3} = \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$z_{4} = \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$x_{2} = - \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
$$x_{3} = \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$x_{4} = \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
$$x^{4} + 81 = 0$$
Because equation degree is equal to = 4 and the free term = -81 < 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:
$$z^{4} = -81$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{4} e^{4 i p} = -81$$
where
$$r = 3$$
- 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:
$$z_{1} = - \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$z_{2} = - \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
$$z_{3} = \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$z_{4} = \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = - \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$x_{2} = - \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
$$x_{3} = \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
$$x_{4} = \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
Rapid solution
[src]
___ ___
3*\/ 2 3*I*\/ 2
x1 = - ------- - ---------
2 2
$$x_{1} = - \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
___ ___
3*\/ 2 3*I*\/ 2
x2 = - ------- + ---------
2 2
$$x_{2} = - \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
___ ___
3*\/ 2 3*I*\/ 2
x3 = ------- - ---------
2 2
$$x_{3} = \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}$$
___ ___
3*\/ 2 3*I*\/ 2
x4 = ------- + ---------
2 2
$$x_{4} = \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}$$
Sum and product of roots
[src]
sum
___ ___ ___ ___ ___ ___ ___ ___
3*\/ 2 3*I*\/ 2 3*\/ 2 3*I*\/ 2 3*\/ 2 3*I*\/ 2 3*\/ 2 3*I*\/ 2
0 + - ------- - --------- + - ------- + --------- + ------- - --------- + ------- + ---------
2 2 2 2 2 2 2 2
$$\left(\left(\frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}\right) - 3 \sqrt{2}\right) + \left(\frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}\right)$$
=
0
$$0$$
product
/ ___ ___\ / ___ ___\ / ___ ___\ / ___ ___\ | 3*\/ 2 3*I*\/ 2 | | 3*\/ 2 3*I*\/ 2 | |3*\/ 2 3*I*\/ 2 | |3*\/ 2 3*I*\/ 2 | 1*|- ------- - ---------|*|- ------- + ---------|*|------- - ---------|*|------- + ---------| \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$1 \left(- \frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}\right) \left(- \frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}\right) \left(\frac{3 \sqrt{2}}{2} - \frac{3 \sqrt{2} i}{2}\right) \left(\frac{3 \sqrt{2}}{2} + \frac{3 \sqrt{2} i}{2}\right)$$
=
81
$$81$$
81
Numerical answer
[src]
x1 = -2.12132034355964 + 2.12132034355964*i
x2 = 2.12132034355964 + 2.12132034355964*i
x3 = -2.12132034355964 - 2.12132034355964*i
x4 = 2.12132034355964 - 2.12132034355964*i
x4 = 2.12132034355964 - 2.12132034355964*i