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
VladimirAG [237]
3 years ago
14

Find all complex numbers z such that z^2=2iplease answer in a+bithank you​

Mathematics
1 answer:
Step2247 [10]3 years ago
3 0

2 Answers:

z = 1 + i   and   z = -1 - i

========================================================

Explanation:

We want z to be a complex number in the form z = a+bi, where a,b are real numbers and i = \sqrt{-1} is imaginary.

Let's plug that into the equation your teacher gave you

z^2 = 2i\\\\(a+bi)^2 = 2i\\\\(a+bi)(a+bi) = 2i\\\\a(a+bi)+bi(a+bi) = 2i\\\\a^2+abi+abi+b^2*i^2 = 2i\\\\a^2+2abi+b^2*(-1) = 2i\\\\a^2+2abi-b^2 = 2i\\\\(a^2-b^2)+2abi = 0+2i\\\\

You could use the FOIL rule to take a shortcut. I'm deciding to be a bit more wordy to show a further breakdown how everything is multiplying out.

Notice that the real part a^2-b^2 must be 0 so that it matches the real part on the right hand side.

a^2-b^2 = 0

(a-b)(a+b) = 0 .... difference of squares rule

a-b = 0 or a+b = 0

a = b or a = -b

So whatever solution z = a+bi is, it must have either a = b or a = -b.

--------------------------------

If a = b, then the 2abi portion on the left side turns into 2a^2*i

Set this equal to 2i on the right hand side and isolate 'a'

2a^2*i = 2i\\\\2a^2 = 2\\\\a^2 = 1\\\\a = 1 \text{ or } a = -1\\\\

So a = 1 leads to b = 1

Or a = -1 leads to b = -1

Two complex solutions so far are:   z = 1 + i   and   z = -1 - i  based on those two cases above.

--------------------------------

Now consider the case that a = -b

We'll effectively have the same steps as the previous section, but the equation to solve now is -2a^2*i = 2i\\\\

The only difference is that negative is out front. You should find that it leads to a^2 = -1, but this has no solutions because we stated earlier that a,b were real numbers.

So if a = -b, then it concludes with a,b being nonreal numbers. Ultimately we rule out the case that a = -b is possible.

Put another way, note how -2a^2 is always negative which clashes with the idea that the right hand side is positive (ignore the 'i' portions). This contradiction means that no real values of 'a' will make the equation -2a^2*i = 2i\\\\ to be true.

--------------------------------

To wrap things up, we only have two solutions and they are  

z = 1 + i   and   z = -1 - i

You can use a tool like WolframAlpha to confirm this.

You might be interested in
Whats the slope of (2,-5),(-6,5)​
Makovka662 [10]
The slope of the line with the points (2,-5) and (-6,5) is 5/-4 (or 10/-8).
3 0
2 years ago
Apple would like to estimate the web browsing battery life of the iPad. The following data represents the battery life, in hours
umka21 [38]

Answer:

12.5 hours

Step-by-step explanation:

The data which represents the battery life, in hours, experienced by six users is given below:

10 14 14 16 13 8

To determine the point estimate for the web browsing battery life of the iPad, we find the mean number of hours.

Mean=\dfrac{10+14+14+16+13+8}{6} \\\\=\dfrac{75}{6} \\\\=12.5

The point estimate for the web browsing battery life of the iPad is: 12.5 hours

8 0
3 years ago
The cost of the studios internet service is given by the formula c=90+30x, where x is the number of months of usage. What is the
blondinia [14]

Answer:

$420

Step-by-step explanation:

C=90+30(11)

11x30=330

330+90=420

6 0
3 years ago
daniel wants to build a sandbox that has a perimeter of 20.5 feet. the length is 5.25 feet. what is the width of their sandbox?
Pavlova-9 [17]
If you subtract 20.5 by 5.25 x 2 you will get 10 by 2 and you have the answer
8 0
1 year ago
Read 2 more answers
Find the area of the figure.
gayaneshka [121]

Answer:

33 units squared is the area of the shape.

Step-by-step explanation:

We know that the the area of the orange (bookmark) shape would be 42 units squared total, since the length is 7 and width is 6 (or vice versa) Now, all we need to find is how many squares inside the rest of the small portion to fill in the shape and subtract that from 42. Adding the squares and half squares in the small portion, I got 9. 42-9 is 33 units squared

3 0
2 years ago
Read 2 more answers
Other questions:
  • What do the factors (x+2)(-8x+70)in the factored form represent?
    5·1 answer
  • Math question please show work for brainliest :)
    5·1 answer
  • -13x - 8 = 5 - 8x -3
    8·1 answer
  • Pls help many thanks
    13·2 answers
  • Write t with magnitude 14 and direction 51 degrees in component form
    14·1 answer
  • Help ASAP please!!!
    13·1 answer
  • I don’t have any of the stuff pls help
    13·1 answer
  • Solve the inequality 5x - 7 > 4x + 8
    7·1 answer
  • What is the value of 2 - (- 2 - 2) - (-2-(-2-2))?​
    6·2 answers
  • Could you please help me with this problem?
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!