Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation sqrt(x-1)=x
-
Equation x^2-6x+8=0
- Equation x^5-32=0
-
Equation (x+6)(2x+6)-20=0
- Express {x} in terms of y where:
- -10*x+18*y=6
- 17*x-19*y=1
- -1*x+19*y=-10
- 4*x+13*y=13
- Derivative of:
-
x^5-32
- Factor polynomial:
- x^5-32
-
Identical expressions
- x^ five - thirty-two = zero
- x to the power of 5 minus 32 equally 0
- x to the power of five minus thirty minus two equally zero
- x5-32=0
- x⁵-32=0
- x^5-32=O
-
Similar expressions
- x^5+32=0
x^5-32=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$x^{5} - 32 = 0$$
Because equation degree is equal to = 5 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 5-th degree of the equation sides:
We get:
$$\sqrt[5]{x^{5}} = \sqrt[5]{32}$$
or
$$x = 2$$
We get the answer: x = 2
All other 4 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{5} = 32$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{5} e^{5 i p} = 32$$
where
$$r = 2$$
- the magnitude of the complex number
Substitute r:
$$e^{5 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = 1$$
so
$$\cos{\left(5 p \right)} = 1$$
and
$$\sin{\left(5 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{5}$$
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:
$$z_{1} = 2$$
$$z_{2} = - \frac{1}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$z_{3} = - \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$z_{4} = - \frac{\sqrt{5}}{2} - \frac{1}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{5} = - \frac{\sqrt{5}}{2} - \frac{1}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = 2$$
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$x_{3} = - \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$x_{4} = - \frac{\sqrt{5}}{2} - \frac{1}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$x_{5} = - \frac{\sqrt{5}}{2} - \frac{1}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$x^{5} - 32 = 0$$
Because equation degree is equal to = 5 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 5-th degree of the equation sides:
We get:
$$\sqrt[5]{x^{5}} = \sqrt[5]{32}$$
or
$$x = 2$$
We get the answer: x = 2
All other 4 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{5} = 32$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{5} e^{5 i p} = 32$$
where
$$r = 2$$
- the magnitude of the complex number
Substitute r:
$$e^{5 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = 1$$
so
$$\cos{\left(5 p \right)} = 1$$
and
$$\sin{\left(5 p \right)} = 0$$
then
$$p = \frac{2 \pi N}{5}$$
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:
$$z_{1} = 2$$
$$z_{2} = - \frac{1}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$z_{3} = - \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$z_{4} = - \frac{\sqrt{5}}{2} - \frac{1}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{5} = - \frac{\sqrt{5}}{2} - \frac{1}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
do backward replacement
$$z = x$$
$$x = z$$
The final answer:
$$x_{1} = 2$$
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$x_{3} = - \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$x_{4} = - \frac{\sqrt{5}}{2} - \frac{1}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$x_{5} = - \frac{\sqrt{5}}{2} - \frac{1}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
Rapid solution
[src]
x1 = 2
$$x_{1} = 2$$
___________
___ / ___
1 \/ 5 / 5 \/ 5
x2 = - - + ----- - 2*I* / - + -----
2 2 \/ 8 8
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
___________
___ / ___
1 \/ 5 / 5 \/ 5
x3 = - - + ----- + 2*I* / - + -----
2 2 \/ 8 8
$$x_{3} = - \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
___________
___ / ___
1 \/ 5 / 5 \/ 5
x4 = - - - ----- - 2*I* / - - -----
2 2 \/ 8 8
$$x_{4} = - \frac{\sqrt{5}}{2} - \frac{1}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
___________
___ / ___
1 \/ 5 / 5 \/ 5
x5 = - - - ----- + 2*I* / - - -----
2 2 \/ 8 8
$$x_{5} = - \frac{\sqrt{5}}{2} - \frac{1}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
x5 = -sqrt(5)/2 - 1/2 + 2*i*sqrt(5/8 - sqrt(5)/8)
Sum and product of roots
[src]
sum
___________ ___________ ___________ ___________
___ / ___ ___ / ___ ___ / ___ ___ / ___
1 \/ 5 / 5 \/ 5 1 \/ 5 / 5 \/ 5 1 \/ 5 / 5 \/ 5 1 \/ 5 / 5 \/ 5
2 + - - + ----- - 2*I* / - + ----- + - - + ----- + 2*I* / - + ----- + - - - ----- - 2*I* / - - ----- + - - - ----- + 2*I* / - - -----
2 2 \/ 8 8 2 2 \/ 8 8 2 2 \/ 8 8 2 2 \/ 8 8
$$\left(\left(- \frac{\sqrt{5}}{2} - \frac{1}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right) + \left(\left(2 + \left(- \frac{1}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right) + \left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right)\right) + \left(- \frac{\sqrt{5}}{2} - \frac{1}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)$$
=
0
$$0$$
product
/ ___________\ / ___________\ / ___________\ / ___________\ | ___ / ___ | | ___ / ___ | | ___ / ___ | | ___ / ___ | | 1 \/ 5 / 5 \/ 5 | | 1 \/ 5 / 5 \/ 5 | | 1 \/ 5 / 5 \/ 5 | | 1 \/ 5 / 5 \/ 5 | 2*|- - + ----- - 2*I* / - + ----- |*|- - + ----- + 2*I* / - + ----- |*|- - - ----- - 2*I* / - - ----- |*|- - - ----- + 2*I* / - - ----- | \ 2 2 \/ 8 8 / \ 2 2 \/ 8 8 / \ 2 2 \/ 8 8 / \ 2 2 \/ 8 8 /
$$2 \left(- \frac{1}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right) \left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right) \left(- \frac{\sqrt{5}}{2} - \frac{1}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right) \left(- \frac{\sqrt{5}}{2} - \frac{1}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)$$
=
32
$$32$$
32
Numerical answer
[src]
x1 = 0.618033988749895 - 1.90211303259031*i
x2 = 2.0
x3 = 0.618033988749895 + 1.90211303259031*i
x4 = -1.61803398874989 - 1.17557050458495*i
x5 = -1.61803398874989 + 1.17557050458495*i
x5 = -1.61803398874989 + 1.17557050458495*i