Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^3=2
-
Equation x*x-x+7=0
-
Equation (x-5)^2-x^2=0
-
Equation (x+7)^3=216
- Express {x} in terms of y where:
- -4*x+7*y=-16
- 12*x-20*y=9
- -1*x-20*y=1
- -5*x-12*y=13
-
Identical expressions
- z^ four - sixteen *i= zero
- z to the power of 4 minus 16 multiply by i equally 0
- z to the power of four minus sixteen multiply by i equally zero
- z4-16*i=0
- z⁴-16*i=0
- z^4-16i=0
- z4-16i=0
- z^4-16*i=O
-
Similar expressions
- z^4+16*i=0
z^4-16*i=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$z^{4} - 16 i = 0$$
Because equation degree is equal to = 4 and the free term = 16*i complex,
so the real solutions of the equation d'not exist
All other 4 root(s) is the complex numbers.
do replacement:
$$w = z$$
then the equation will be the:
$$w^{4} = 16 i$$
Any complex number can presented so:
$$w = r e^{i p}$$
substitute to the equation
$$r^{4} e^{4 i p} = 16 i$$
where
$$r = 2$$
- the magnitude of the complex number
Substitute r:
$$e^{4 i p} = i$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = i$$
so
$$\cos{\left(4 p \right)} = 0$$
and
$$\sin{\left(4 p \right)} = 1$$
then
$$p = \frac{\pi N}{2} + \frac{\pi}{8}$$
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:
$$w_{1} = - 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$w_{2} = 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$w_{3} = - 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$w_{4} = 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
do backward replacement
$$w = z$$
$$z = w$$
The final answer:
$$z_{1} = - 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{2} = 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{3} = - 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$z_{4} = 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$z^{4} - 16 i = 0$$
Because equation degree is equal to = 4 and the free term = 16*i complex,
so the real solutions of the equation d'not exist
All other 4 root(s) is the complex numbers.
do replacement:
$$w = z$$
then the equation will be the:
$$w^{4} = 16 i$$
Any complex number can presented so:
$$w = r e^{i p}$$
substitute to the equation
$$r^{4} e^{4 i p} = 16 i$$
where
$$r = 2$$
- the magnitude of the complex number
Substitute r:
$$e^{4 i p} = i$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = i$$
so
$$\cos{\left(4 p \right)} = 0$$
and
$$\sin{\left(4 p \right)} = 1$$
then
$$p = \frac{\pi N}{2} + \frac{\pi}{8}$$
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:
$$w_{1} = - 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$w_{2} = 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$w_{3} = - 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$w_{4} = 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
do backward replacement
$$w = z$$
$$z = w$$
The final answer:
$$z_{1} = - 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{2} = 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
$$z_{3} = - 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
$$z_{4} = 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
The graph
Rapid solution
[src]
___________ ___________
/ ___ / ___
/ 1 \/ 2 / 1 \/ 2
z1 = - 2* / - - ----- + 2*I* / - + -----
\/ 2 4 \/ 2 4
$$z_{1} = - 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
___________ ___________
/ ___ / ___
/ 1 \/ 2 / 1 \/ 2
z2 = 2* / - - ----- - 2*I* / - + -----
\/ 2 4 \/ 2 4
$$z_{2} = 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}$$
___________ ___________
/ ___ / ___
/ 1 \/ 2 / 1 \/ 2
z3 = - 2* / - + ----- - 2*I* / - - -----
\/ 2 4 \/ 2 4
$$z_{3} = - 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
___________ ___________
/ ___ / ___
/ 1 \/ 2 / 1 \/ 2
z4 = 2* / - + ----- + 2*I* / - - -----
\/ 2 4 \/ 2 4
$$z_{4} = 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}$$
z4 = 2*sqrt(sqrt(2)/4 + 1/2) + 2*i*sqrt(1/2 - sqrt(2)/4)
Sum and product of roots
[src]
sum
___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________
/ ___ / ___ / ___ / ___ / ___ / ___ / ___ / ___
/ 1 \/ 2 / 1 \/ 2 / 1 \/ 2 / 1 \/ 2 / 1 \/ 2 / 1 \/ 2 / 1 \/ 2 / 1 \/ 2
- 2* / - - ----- + 2*I* / - + ----- + 2* / - - ----- - 2*I* / - + ----- + - 2* / - + ----- - 2*I* / - - ----- + 2* / - + ----- + 2*I* / - - -----
\/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4
$$\left(\left(- 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) + \left(\left(2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) + \left(- 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right)\right) + \left(2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)$$
=
0
$$0$$
product
/ ___________ ___________\ / ___________ ___________\ / ___________ ___________\ / ___________ ___________\ | / ___ / ___ | | / ___ / ___ | | / ___ / ___ | | / ___ / ___ | | / 1 \/ 2 / 1 \/ 2 | | / 1 \/ 2 / 1 \/ 2 | | / 1 \/ 2 / 1 \/ 2 | | / 1 \/ 2 / 1 \/ 2 | |- 2* / - - ----- + 2*I* / - + ----- |*|2* / - - ----- - 2*I* / - + ----- |*|- 2* / - + ----- - 2*I* / - - ----- |*|2* / - + ----- + 2*I* / - - ----- | \ \/ 2 4 \/ 2 4 / \ \/ 2 4 \/ 2 4 / \ \/ 2 4 \/ 2 4 / \ \/ 2 4 \/ 2 4 /
$$\left(- 2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(2 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 2 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(- 2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 2 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)$$
=
-16*I
$$- 16 i$$
-16*i
Numerical answer
[src]
z1 = -1.84775906502257 - 0.76536686473018*i
z2 = -0.76536686473018 + 1.84775906502257*i
z3 = 0.76536686473018 - 1.84775906502257*i
z4 = 1.84775906502257 + 0.76536686473018*i
z4 = 1.84775906502257 + 0.76536686473018*i