1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
mixer [17]
3 years ago
9

Assume z = x + iy, then find a complex number z satisfying the given equation. d. 2z8 – 2z4 + 1 = 0

Mathematics
1 answer:
kodGreya [7K]3 years ago
3 0

Answer: complex equations has n number of solutions, been n the equation degree. In this case:

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i11,25°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i101,25°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i191,25°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i281,25°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i78,75°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i168,75°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i258,75°}

Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i348,75°}

Step-by-step explanation:

I start with a variable substitution:

Z^{4} = X

Then:

2X^{2}-2X+1=0

Solving the quadratic equation:

X_{1} =\frac{2+\sqrt{4-4*2*1} }{2*2} \\X_{2} =\frac{2-\sqrt{4-4*2*1} }{2*2}

X=\left \{ {{0,5+0,5i} \atop {0,5-0,5i}} \right.

Replacing for the original variable:

Z=\sqrt[4]{0,5+0,5i}

or Z=\sqrt[4]{0,5-0,5i}

Remembering that complex numbers can be written as:

Z=a+ib=|Z|e^{ic}

Using this:

Z=\left \{ {{{\frac{\sqrt{2}}{2} e^{i45°} } \atop {{\frac{\sqrt{2}}{2} e^{i-45°} }} \right.

Solving for the modulus and the angle:

Z=\left \{ {{\sqrt[4]{\frac{\sqrt{2}}{2} e^{i45}} = \sqrt[4]{\frac{\sqrt{2}}{2} } \sqrt[4]{e^{i45}} } \atop {\sqrt[4]{\frac{\sqrt{2}}{2} e^{i-45}} = \sqrt[4]{\frac{\sqrt{2}}{2} } \sqrt[4]{e^{i-45}} }} \right.

The possible angle respond to:

RAng_{12...n} =\frac{Ang +360*(i-1)}{n}

Been "RAng" the resultant angle, "Ang" the original angle, "n" the degree of the root and "i" a value between 1 and "n"

In this case n=4 with 2 different angles: Ang = 45º and Ang = 315º

Obtaining 8 different angles, therefore 8 different solutions.

You might be interested in
What would be the most logical first step for solving this quadratic equation?
Sav [38]

Answer:

D. Subtract 4 from both sides

Explanation:

I just learned about quadratic equations in school. The first step of solving quadratic equations is to make sure it is in standard form and equal to 0.

Standard form:

ax^2 + bx + c = 0

4 0
2 years ago
A test has 20 questions. If peter gets 80% correct, how many questions did peter missed?
Umnica [9.8K]
16 because if you put 16 into a percentage it's 80. 16 divide 20 times 100 equal 80
8 0
3 years ago
Read 2 more answers
PLEASE HELP 2O POINTS
vovangra [49]
C. The first is a solution, but the second is not.

5x - y/3 = 13 ; (2,-9)
substitute the letters:
5(2) - (-9/3) = 13 
10 - (-3) = 13 : note that deducting a number with a negative sign turns both sign as positive.
10 + 3 = 13 ;
13 = 13

5x - y/3 = 13 ; (3,-6)
5(3) - (-6/3) = 13
15 - (-2) = 13
15 + 2 = 13
17 = 13 not equal. not a solution

hope i could help
4 0
3 years ago
Read 2 more answers
3) At a school several teachers were holding a contest to see which class could earn the most trivia points. Mrs. Clawson's clas
Snezhnost [94]

Answer:

Mean: 89

Median: 90

Mode: 94

Range: 12

5 0
2 years ago
Factor completely, then place the factors in the proper location on the grid. 25a2 +9b2 + 30ab
sergij07 [2.7K]

Answer:

(5a+3b)(5a+3b) or SQ(5a+3b)

Step-by-step explanation:

Now you can plot by referring to the above factors

8 0
3 years ago
Other questions:
  • Given the function, f(x)=x+1+2, choose the correct transformation.
    8·1 answer
  • Only number 8 plsssss bet no one can get this
    11·1 answer
  • I need help asap! <br> What is <br> -3^6-2<br> -3^6-1<br> -3^6-0<br> -3^6+1<br> -3^6+2
    5·1 answer
  • Find a number lower than 19 that has more factors than 19,21,23, and 25
    7·1 answer
  • What is the answer 2-2n=3n+17
    15·2 answers
  • The standard error of the estimate is a type of measure of _____.
    9·1 answer
  • A female lion at the zoo what is 190 pounds more than the female cheetah. Let c represent the weight in pounds of the cheetah. W
    14·1 answer
  • HELP AND I WILL GIVE BRANILIEST
    13·1 answer
  • Y varies inversely with x: If y = 10 while x = 3, find y when x = 6.
    9·1 answer
  • Jeremy bakes two pans of brownies that are the same size. One pan has nuts in it and the other pan does not. He cuts the pan wit
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!