Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 3x^2=2x+x^3
- Equation x^3+2*x^2-36*x-72=0
-
Equation (x+2)^5=-32
-
Equation (x+10)2=(5-x)2
- Express {x} in terms of y where:
- 4*x+11*y=9
- -20*x-1*y=3
- -14*x+14*y=6
- 5*x+15*y=7
- Limit of the function:
-
-32
-
Identical expressions
- (x+ two)^ five =- thirty-two
- (x plus 2) to the power of 5 equally minus 32
- (x plus two) to the power of five equally minus thirty minus two
- (x+2)5=-32
- x+25=-32
- (x+2)⁵=-32
- x+2^5=-32
-
Similar expressions
- (x+2)^5=+32
- ((x-3)^2)/7+((x-5)^2)/14=((x-7)^2)/21+(x-19)/70
- (x-3)^2+(3-x)*(x+3)=(x+2)^2-x^2
- (x-2)^5=-32
- log2^2(x-3)^2-14log2(6-2x)+30=0
(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
$$x + 2 = 2 \sqrt[5]{-1}$$
Expand brackets in the right part
Move free summands (without x)
from left part to right part, we given:
$$x = -2 + 2 \sqrt[5]{-1}$$
We get the answer: x = -2 + 2*(-1)^(1/5)
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
$$p = \frac{2 \pi N}{5} + \frac{\pi}{5}$$
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:
$$z_{1} = -2$$
$$z_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{3} = - \frac{\sqrt{5}}{2} + \frac{1}{2} - \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{4} = \frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$z_{5} = - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
do backward replacement
$$z = x + 2$$
$$x = z - 2$$
The final answer:
$$x_{1} = -4$$
$$x_{2} = - \frac{3}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$x_{3} = - \frac{3}{2} - \frac{\sqrt{5}}{2} - \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$x_{4} = - \frac{3}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$x_{5} = - \frac{3}{2} - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
$$\left(x + 2\right)^{5} = -32$$
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
$$x + 2 = 2 \sqrt[5]{-1}$$
Expand brackets in the right part
2 + x = -2*1^1/5
Move free summands (without x)
from left part to right part, we given:
$$x = -2 + 2 \sqrt[5]{-1}$$
We get the answer: x = -2 + 2*(-1)^(1/5)
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
$$p = \frac{2 \pi N}{5} + \frac{\pi}{5}$$
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:
$$z_{1} = -2$$
$$z_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{3} = - \frac{\sqrt{5}}{2} + \frac{1}{2} - \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{4} = \frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$z_{5} = - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
do backward replacement
$$z = x + 2$$
$$x = z - 2$$
The final answer:
$$x_{1} = -4$$
$$x_{2} = - \frac{3}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$x_{3} = - \frac{3}{2} - \frac{\sqrt{5}}{2} - \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$x_{4} = - \frac{3}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$x_{5} = - \frac{3}{2} - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
Rapid solution
[src]
x1 = -4
$$x_{1} = -4$$
___________
___ / ___
3 \/ 5 / 5 \/ 5
x2 = - - - ----- - 2*I* / - + -----
2 2 \/ 8 8
$$x_{2} = - \frac{3}{2} - \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
___________
___ / ___
3 \/ 5 / 5 \/ 5
x3 = - - - ----- + 2*I* / - + -----
2 2 \/ 8 8
$$x_{3} = - \frac{3}{2} - \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}$$
___________
___ / ___
3 \/ 5 / 5 \/ 5
x4 = - - + ----- - 2*I* / - - -----
2 2 \/ 8 8
$$x_{4} = - \frac{3}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
___________
___ / ___
3 \/ 5 / 5 \/ 5
x5 = - - + ----- + 2*I* / - - -----
2 2 \/ 8 8
$$x_{5} = - \frac{3}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
x5 = -3/2 + sqrt(5)/2 + 2*i*sqrt(5/8 - sqrt(5)/8)
Sum and product of roots
[src]
sum
___________ ___________ ___________ ___________
___ / ___ ___ / ___ ___ / ___ ___ / ___
3 \/ 5 / 5 \/ 5 3 \/ 5 / 5 \/ 5 3 \/ 5 / 5 \/ 5 3 \/ 5 / 5 \/ 5
-4 + - - - ----- - 2*I* / - + ----- + - - - ----- + 2*I* / - + ----- + - - + ----- - 2*I* / - - ----- + - - + ----- + 2*I* / - - -----
2 2 \/ 8 8 2 2 \/ 8 8 2 2 \/ 8 8 2 2 \/ 8 8
$$\left(\left(- \frac{3}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right) + \left(\left(-4 + \left(- \frac{3}{2} - \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right) + \left(- \frac{3}{2} - \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right)\right) + \left(- \frac{3}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)$$
=
-10
$$-10$$
product
/ ___________\ / ___________\ / ___________\ / ___________\ | ___ / ___ | | ___ / ___ | | ___ / ___ | | ___ / ___ | | 3 \/ 5 / 5 \/ 5 | | 3 \/ 5 / 5 \/ 5 | | 3 \/ 5 / 5 \/ 5 | | 3 \/ 5 / 5 \/ 5 | -4*|- - - ----- - 2*I* / - + ----- |*|- - - ----- + 2*I* / - + ----- |*|- - + ----- - 2*I* / - - ----- |*|- - + ----- + 2*I* / - - ----- | \ 2 2 \/ 8 8 / \ 2 2 \/ 8 8 / \ 2 2 \/ 8 8 / \ 2 2 \/ 8 8 /
$$- 4 \left(- \frac{3}{2} - \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right) \left(- \frac{3}{2} - \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right) \left(- \frac{3}{2} + \frac{\sqrt{5}}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right) \left(- \frac{3}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)$$
=
-64
$$-64$$
-64
Numerical answer
[src]
x1 = -0.381966011250105 - 1.17557050458495*i
x2 = -2.61803398874989 - 1.90211303259031*i
x3 = -2.61803398874989 + 1.90211303259031*i
x4 = -0.381966011250105 + 1.17557050458495*i
x5 = -4.0
x5 = -4.0
The graph