Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation z^5+32=0
- Equation x⁴-3x²-4=0
- Equation 237*x=158+142
- Equation x*200=5440+360
- Express {x} in terms of y where:
- 60x^2y+75x^2y^2+120xy^260x2y+75x2y2+120xy2приx=-0,5x=−0,5,y=0,4y=0,4.
- 5*x-5*y=20
- 9*x+18*y=6
- 15*x-15*y=16
-
Identical expressions
- z^ five + thirty-two = zero
- z to the power of 5 plus 32 equally 0
- z to the power of five plus thirty minus two equally zero
- z5+32=0
- z⁵+32=0
- z^5+32=O
-
Similar expressions
- z^5-32=0
z^5+32=0 equation
The solution
Detail solution
Given the equation
$$z^{5} + 32 = 0$$
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:
$$\sqrt[5]{z^{5}} = \sqrt[5]{-32}$$
or
$$z = 2 \sqrt[5]{-1}$$
Expand brackets in the right part
We get the answer: z = 2*(-1)^(1/5)
All other 4 root(s) is the complex numbers.
do replacement:
$$w = z$$
then the equation will be the:
$$w^{5} = -32$$
Any complex number can presented so:
$$w = r e^{i p}$$
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 w
Consequently, the solution will be for w:
$$w_{1} = -2$$
$$w_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$w_{3} = - \frac{\sqrt{5}}{2} + \frac{1}{2} - \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$w_{4} = \frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$w_{5} = - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
do backward replacement
$$w = z$$
$$z = w$$
The final answer:
$$z_{1} = -2$$
$$z_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{3} = - \frac{\sqrt{5}}{2} + \frac{1}{2} - \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{4} = \frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$z_{5} = - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
$$z^{5} + 32 = 0$$
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:
$$\sqrt[5]{z^{5}} = \sqrt[5]{-32}$$
or
$$z = 2 \sqrt[5]{-1}$$
Expand brackets in the right part
z = -2*1^1/5
We get the answer: z = 2*(-1)^(1/5)
All other 4 root(s) is the complex numbers.
do replacement:
$$w = z$$
then the equation will be the:
$$w^{5} = -32$$
Any complex number can presented so:
$$w = r e^{i p}$$
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 w
Consequently, the solution will be for w:
$$w_{1} = -2$$
$$w_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$w_{3} = - \frac{\sqrt{5}}{2} + \frac{1}{2} - \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$w_{4} = \frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$w_{5} = - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
do backward replacement
$$w = z$$
$$z = w$$
The final answer:
$$z_{1} = -2$$
$$z_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{3} = - \frac{\sqrt{5}}{2} + \frac{1}{2} - \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
$$z_{4} = \frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}$$
$$z_{5} = - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}$$
Sum and product of roots
[src]
sum
/ ___________ ___________ ___________ ___________\ / ___________ ___________ ___________ ___________\
| / ___ / ___ / ___ / ___ | | / ___ / ___ / ___ / ___ |
/ ___________ ___________\ | / 5 \/ 5 / 5 \/ 5 ___ / 5 \/ 5 ___ / 5 \/ 5 | ___________ ___________ | / 5 \/ 5 / 5 \/ 5 ___ / 5 \/ 5 ___ / 5 \/ 5 | ___________ ___________ ___________
___ | / ___ / ___ | | / - - ----- / - + ----- \/ 5 * / - - ----- \/ 5 * / - + ----- | / ___ / ___ | / - + ----- / - - ----- \/ 5 * / - - ----- \/ 5 * / - + ----- | / ___ / ___ ___ / ___
1 \/ 5 | / 5 \/ 5 ___ / 5 \/ 5 | 1 | \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 | / 5 \/ 5 / 5 \/ 5 1 |\/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 | / 5 \/ 5 / 5 \/ 5 1 \/ 5 / 5 \/ 5
-2 + - - ----- + I*|- / - - ----- - \/ 5 * / - - ----- | + - + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2* / - - ----- * / - + ----- + - + I*|---------------- - ---------------- + ---------------------- + ----------------------| - 2* / - - ----- * / - + ----- + - + ----- + 2*I* / - - -----
2 2 \ \/ 8 8 \/ 8 8 / 2 \ 2 2 2 2 / \/ 8 8 \/ 8 8 2 \ 2 2 2 2 / \/ 8 8 \/ 8 8 2 2 \/ 8 8
$$\left(\left(\left(-2 + \left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + i \left(- \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right)\right) + \left(\frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)\right)\right) + \left(- 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)$$
=
/ ___________ ___________ ___________ ___________\ / ___________ ___________ ___________ ___________\
| / ___ / ___ / ___ / ___ | | / ___ / ___ / ___ / ___ |
/ ___________ ___________\ | / 5 \/ 5 / 5 \/ 5 ___ / 5 \/ 5 ___ / 5 \/ 5 | | / 5 \/ 5 / 5 \/ 5 ___ / 5 \/ 5 ___ / 5 \/ 5 | ___________
| / ___ / ___ | | / - + ----- / - - ----- \/ 5 * / - - ----- \/ 5 * / - + ----- | | / - - ----- / - + ----- \/ 5 * / - - ----- \/ 5 * / - + ----- | / ___
| / 5 \/ 5 ___ / 5 \/ 5 | |\/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 | | \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 | / 5 \/ 5
I*|- / - - ----- - \/ 5 * / - - ----- | + I*|---------------- - ---------------- + ---------------------- + ----------------------| + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2*I* / - - -----
\ \/ 8 8 \/ 8 8 / \ 2 2 2 2 / \ 2 2 2 2 / \/ 8 8
$$i \left(- \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right) + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)$$
product
/ / ___________ ___________ ___________ ___________\ \ / / ___________ ___________ ___________ ___________\ \
| | / ___ / ___ / ___ / ___ | | | | / ___ / ___ / ___ / ___ | |
/ / ___________ ___________\\ | | / 5 \/ 5 / 5 \/ 5 ___ / 5 \/ 5 ___ / 5 \/ 5 | ___________ ___________| | | / 5 \/ 5 / 5 \/ 5 ___ / 5 \/ 5 ___ / 5 \/ 5 | ___________ ___________| / ___________\
| ___ | / ___ / ___ || | | / - - ----- / - + ----- \/ 5 * / - - ----- \/ 5 * / - + ----- | / ___ / ___ | | | / - + ----- / - - ----- \/ 5 * / - - ----- \/ 5 * / - + ----- | / ___ / ___ | | ___ / ___ |
|1 \/ 5 | / 5 \/ 5 ___ / 5 \/ 5 || |1 | \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 | / 5 \/ 5 / 5 \/ 5 | |1 |\/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 | / 5 \/ 5 / 5 \/ 5 | |1 \/ 5 / 5 \/ 5 |
-2*|- - ----- + I*|- / - - ----- - \/ 5 * / - - ----- ||*|- + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2* / - - ----- * / - + ----- |*|- + I*|---------------- - ---------------- + ---------------------- + ----------------------| - 2* / - - ----- * / - + ----- |*|- + ----- + 2*I* / - - ----- |
\2 2 \ \/ 8 8 \/ 8 8 // \2 \ 2 2 2 2 / \/ 8 8 \/ 8 8 / \2 \ 2 2 2 2 / \/ 8 8 \/ 8 8 / \2 2 \/ 8 8 /
$$\left(\frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)\right) \left(- 2 \left(- \frac{\sqrt{5}}{2} + \frac{1}{2} + i \left(- \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right)\right) \left(- 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right) \left(\frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)$$
=
___________ ___________ ____________ ___________ ____________ ___________
____________ ____________ ___________ ____ / ___ ___________ ___ / ___ / ___ / ___ ___ / ___ / ___
___ ___ / ___ / ___ / ___ I*\/ 10 *\/ 5 - \/ 5 ____ / ___ 3*I*\/ 2 *\/ 5 - \/ 5 5*I*\/ -5 + \/ 5 *\/ 5 + \/ 5 3*I*\/ 5 *\/ -5 + \/ 5 *\/ 5 + \/ 5
-12 + 4*\/ 5 + 4*\/ 2 *\/ -5 + \/ 5 + I*\/ -5 + \/ 5 *\/ 5 - \/ 5 + ----------------------- - I*\/ 10 *\/ 5 + \/ 5 - ------------------------ + ---------------------------------- + ----------------------------------------
2 2 2 2
$$\frac{3 \sqrt{5} i \sqrt{-5 + \sqrt{5}} \sqrt{\sqrt{5} + 5}}{2} - 12 + \frac{5 i \sqrt{-5 + \sqrt{5}} \sqrt{\sqrt{5} + 5}}{2} + i \sqrt{-5 + \sqrt{5}} \sqrt{5 - \sqrt{5}} + 4 \sqrt{5} - \sqrt{10} i \sqrt{\sqrt{5} + 5} - \frac{3 \sqrt{2} i \sqrt{5 - \sqrt{5}}}{2} + \frac{\sqrt{10} i \sqrt{5 - \sqrt{5}}}{2} + 4 \sqrt{2} \sqrt{-5 + \sqrt{5}}$$
-12 + 4*sqrt(5) + 4*sqrt(2)*sqrt(-5 + sqrt(5)) + i*sqrt(-5 + sqrt(5))*sqrt(5 - sqrt(5)) + i*sqrt(10)*sqrt(5 - sqrt(5))/2 - i*sqrt(10)*sqrt(5 + sqrt(5)) - 3*i*sqrt(2)*sqrt(5 - sqrt(5))/2 + 5*i*sqrt(-5 + sqrt(5))*sqrt(5 + sqrt(5))/2 + 3*i*sqrt(5)*sqrt(-5 + sqrt(5))*sqrt(5 + sqrt(5))/2
Rapid solution
[src]
z1 = -2
$$z_{1} = -2$$
/ ___________ ___________\
___ | / ___ / ___ |
1 \/ 5 | / 5 \/ 5 ___ / 5 \/ 5 |
z2 = - - ----- + I*|- / - - ----- - \/ 5 * / - - ----- |
2 2 \ \/ 8 8 \/ 8 8 /
$$z_{2} = - \frac{\sqrt{5}}{2} + \frac{1}{2} + i \left(- \sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)$$
/ ___________ ___________ ___________ ___________\
| / ___ / ___ / ___ / ___ |
| / 5 \/ 5 / 5 \/ 5 ___ / 5 \/ 5 ___ / 5 \/ 5 | ___________ ___________
| / - - ----- / - + ----- \/ 5 * / - - ----- \/ 5 * / - + ----- | / ___ / ___
1 | \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 | / 5 \/ 5 / 5 \/ 5
z3 = - + I*|- ---------------- - ---------------- + ---------------------- - ----------------------| + 2* / - - ----- * / - + -----
2 \ 2 2 2 2 / \/ 8 8 \/ 8 8
$$z_{3} = \frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)$$
/ ___________ ___________ ___________ ___________\
| / ___ / ___ / ___ / ___ |
| / 5 \/ 5 / 5 \/ 5 ___ / 5 \/ 5 ___ / 5 \/ 5 | ___________ ___________
| / - + ----- / - - ----- \/ 5 * / - - ----- \/ 5 * / - + ----- | / ___ / ___
1 |\/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 | / 5 \/ 5 / 5 \/ 5
z4 = - + I*|---------------- - ---------------- + ---------------------- + ----------------------| - 2* / - - ----- * / - + -----
2 \ 2 2 2 2 / \/ 8 8 \/ 8 8
$$z_{4} = - 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{2} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)$$
___________
___ / ___
1 \/ 5 / 5 \/ 5
z5 = - + ----- + 2*I* / - - -----
2 2 \/ 8 8
$$z_{5} = \frac{1}{2} + \frac{\sqrt{5}}{2} + 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$$
z5 = 1/2 + sqrt(5)/2 + 2*i*sqrt(5/8 - sqrt(5)/8)
Numerical answer
[src]
z1 = 1.61803398874989 + 1.17557050458495*i
z2 = -2.0
z3 = 1.61803398874989 - 1.17557050458495*i
z4 = -0.618033988749895 - 1.90211303259031*i
z5 = -0.618033988749895 + 1.90211303259031*i
z5 = -0.618033988749895 + 1.90211303259031*i