Mister Exam
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How to use it?
- Equation:
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Equation 5x-2=1
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Equation (x-1)(x^2+6x+9)=5(x+3)
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Equation x^2+2*x+10=0
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Identical expressions
- z^ three -i= zero
- z cubed minus i equally 0
- z to the power of three minus i equally zero
- z3-i=0
- z³-i=0
- z to the power of 3-i=0
- z^3-i=O
-
Similar expressions
- z^3+i=0
z^3-i=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$z^{3} - i = 0$$
Because equation degree is equal to = 3 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 3-th degree of the equation sides:
We get:
$$\sqrt[3]{z^{3}} = \sqrt[3]{i}$$
or
$$z = \sqrt[3]{i}$$
Expand brackets in the right part
We get the answer: z = i^(1/3)
All other 3 root(s) is the complex numbers.
do replacement:
$$w = z$$
then the equation will be the:
$$w^{3} = i$$
Any complex number can presented so:
$$w = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = i$$
where
$$r = 1$$
- the magnitude of the complex number
Substitute r:
$$e^{3 i p} = i$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = i$$
so
$$\cos{\left(3 p \right)} = 0$$
and
$$\sin{\left(3 p \right)} = 1$$
then
$$p = \frac{2 \pi N}{3} + \frac{\pi}{6}$$
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:
$$w_{1} = - i$$
$$w_{2} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$
$$w_{3} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
do backward replacement
$$w = z$$
$$z = w$$
The final answer:
$$z_{1} = - i$$
$$z_{2} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$
$$z_{3} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
$$z^{3} - i = 0$$
Because equation degree is equal to = 3 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 3-th degree of the equation sides:
We get:
$$\sqrt[3]{z^{3}} = \sqrt[3]{i}$$
or
$$z = \sqrt[3]{i}$$
Expand brackets in the right part
z = i^1/3
We get the answer: z = i^(1/3)
All other 3 root(s) is the complex numbers.
do replacement:
$$w = z$$
then the equation will be the:
$$w^{3} = i$$
Any complex number can presented so:
$$w = r e^{i p}$$
substitute to the equation
$$r^{3} e^{3 i p} = i$$
where
$$r = 1$$
- the magnitude of the complex number
Substitute r:
$$e^{3 i p} = i$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = i$$
so
$$\cos{\left(3 p \right)} = 0$$
and
$$\sin{\left(3 p \right)} = 1$$
then
$$p = \frac{2 \pi N}{3} + \frac{\pi}{6}$$
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:
$$w_{1} = - i$$
$$w_{2} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$
$$w_{3} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
do backward replacement
$$w = z$$
$$z = w$$
The final answer:
$$z_{1} = - i$$
$$z_{2} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$
$$z_{3} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
Vieta's Theorem
it is reduced cubic equation
$$p z^{2} + q z + v + z^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = - i$$
Vieta Formulas
$$z_{1} + z_{2} + z_{3} = - p$$
$$z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3} = q$$
$$z_{1} z_{2} z_{3} = v$$
$$z_{1} + z_{2} + z_{3} = 0$$
$$z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3} = 0$$
$$z_{1} z_{2} z_{3} = - i$$
$$p z^{2} + q z + v + z^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = - i$$
Vieta Formulas
$$z_{1} + z_{2} + z_{3} = - p$$
$$z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3} = q$$
$$z_{1} z_{2} z_{3} = v$$
$$z_{1} + z_{2} + z_{3} = 0$$
$$z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3} = 0$$
$$z_{1} z_{2} z_{3} = - i$$
The graph
Rapid solution
[src]
z1 = -I
$$z_{1} = - i$$
___
I \/ 3
z2 = - - -----
2 2
$$z_{2} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$
___
I \/ 3
z3 = - + -----
2 2
$$z_{3} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
z3 = sqrt(3)/2 + i/2
Sum and product of roots
[src]
sum
___ ___
I \/ 3 I \/ 3
-I + - - ----- + - + -----
2 2 2 2
$$\left(- i + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)\right) + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)$$
=
0
$$0$$
product
/ ___\ / ___\ |I \/ 3 | |I \/ 3 | -I*|- - -----|*|- + -----| \2 2 / \2 2 /
$$- i \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right) \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)$$
=
I
$$i$$
i
Numerical answer
[src]
z1 = -0.866025403784439 + 0.5*i
z2 = 0.866025403784439 + 0.5*i
z3 = -1.0*i
z3 = -1.0*i