Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 2*sin(x)-1=0
-
Equation x^2-6x+5=0
-
Equation x^2=-49
-
Equation -33-y=-19
- Express {x} in terms of y where:
- -18*x+9*y=-15
- 15*x-3*y=-8
- -2*x-19*y=3
- 4*x-7*y=12
- Canonical form:
- 32
- Expression:
- 32
- 32
-
Identical expressions
- (x+ two)^ five = thirty-two
- (x plus 2) to the power of 5 equally 32
- (x plus two) to the power of five equally thirty minus two
- (x+2)5=32
- x+25=32
- (x+2)⁵=32
- x+2^5=32
-
Similar expressions
- sqrt(16+(2x-3)^2)=4-(cos(5pix/3))^2
- (x-2)^5=32
- (-5*x-3)*(2*x-1)-a=0
- ((5x+1)/(2x-3))^2+((3-2x)/(5x-1))^2=82/9
(x+2)^5=32 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\left(x + 2\right)^{5} = 32$$
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:
$$\sqrt[5]{\left(x + 2\right)^{5}} = \sqrt[5]{32}$$
or
$$x + 2 = 2$$
Move free summands (without x)
from left part to right part, we given:
$$x = 0$$
We get the answer: x = 0
All other 4 root(s) is the complex numbers.
do replacement:
$$z = x + 2$$
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
$$z = x + 2$$
$$x = z - 2$$
The final answer:
$$x_{1} = 0$$
$$x_{2} = - \frac{5}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$x_{3} = - \frac{5}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$x_{4} = - \frac{5}{2} - \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$x_{5} = - \frac{5}{2} - \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$\left(x + 2\right)^{5} = 32$$
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:
$$\sqrt[5]{\left(x + 2\right)^{5}} = \sqrt[5]{32}$$
or
$$x + 2 = 2$$
Move free summands (without x)
from left part to right part, we given:
$$x = 0$$
We get the answer: x = 0
All other 4 root(s) is the complex numbers.
do replacement:
$$z = x + 2$$
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
$$z = x + 2$$
$$x = z - 2$$
The final answer:
$$x_{1} = 0$$
$$x_{2} = - \frac{5}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$x_{3} = - \frac{5}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
$$x_{4} = - \frac{5}{2} - \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$x_{5} = - \frac{5}{2} - \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
___________
___ / ___
5 \/ 5 / 5 \/ 5
x2 = - - - ----- - 2*I* / - - -----
2 2 \/ 8 8
$$x_{2} = - \frac{5}{2} - \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
___________
___ / ___
5 \/ 5 / 5 \/ 5
x3 = - - - ----- + 2*I* / - - -----
2 2 \/ 8 8
$$x_{3} = - \frac{5}{2} - \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
___________
___ / ___
5 \/ 5 / 5 \/ 5
x4 = - - + ----- - 2*I* / - + -----
2 2 \/ 8 8
$$x_{4} = - \frac{5}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
___________
___ / ___
5 \/ 5 / 5 \/ 5
x5 = - - + ----- + 2*I* / - + -----
2 2 \/ 8 8
$$x_{5} = - \frac{5}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
x5 = -5/2 + sqrt(5)/2 + 2*i*sqrt(sqrt(5)/8 + 5/8)
Sum and product of roots
[src]
sum
___________ ___________ ___________ ___________
___ / ___ ___ / ___ ___ / ___ ___ / ___
5 \/ 5 / 5 \/ 5 5 \/ 5 / 5 \/ 5 5 \/ 5 / 5 \/ 5 5 \/ 5 / 5 \/ 5
- - - ----- - 2*I* / - - ----- + - - - ----- + 2*I* / - - ----- + - - + ----- - 2*I* / - + ----- + - - + ----- + 2*I* / - + -----
2 2 \/ 8 8 2 2 \/ 8 8 2 2 \/ 8 8 2 2 \/ 8 8
$$\left(\left(- \frac{5}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right) + \left(\left(- \frac{5}{2} - \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right) + \left(- \frac{5}{2} - \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right)\right) + \left(- \frac{5}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)$$
=
-10
$$-10$$
product
/ ___________\ / ___________\ / ___________\ / ___________\ | ___ / ___ | | ___ / ___ | | ___ / ___ | | ___ / ___ | | 5 \/ 5 / 5 \/ 5 | | 5 \/ 5 / 5 \/ 5 | | 5 \/ 5 / 5 \/ 5 | | 5 \/ 5 / 5 \/ 5 | 0*|- - - ----- - 2*I* / - - ----- |*|- - - ----- + 2*I* / - - ----- |*|- - + ----- - 2*I* / - + ----- |*|- - + ----- + 2*I* / - + ----- | \ 2 2 \/ 8 8 / \ 2 2 \/ 8 8 / \ 2 2 \/ 8 8 / \ 2 2 \/ 8 8 /
$$0 \left(- \frac{5}{2} - \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right) \left(- \frac{5}{2} - \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right) \left(- \frac{5}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right) \left(- \frac{5}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = -1.38196601125011 - 1.90211303259031*i
x2 = -3.61803398874989 - 1.17557050458495*i
x3 = -1.38196601125011 + 1.90211303259031*i
x4 = 0.0
x5 = -3.61803398874989 + 1.17557050458495*i
x5 = -3.61803398874989 + 1.17557050458495*i
The graph