Mathematics High School

If (x+iy)³ = y+vi then show that (y/x + v/y) = 4(x²-y²)​

Answers

Answer 1

Answer:

Step-by-step explanation:

If (x+iy) = Y+vi Then Show That (y/x + V/y) = 4(x-y)
Answer 2

[tex] \large\underline{\sf{Solution-}}[/tex]

Given that,

[tex]\rm \longmapsto\: {(x + iy)}^{3} = y + iv[/tex]

[tex]\rm \longmapsto\: {x}^{3} + 3 {x}^{2}(iy) + 3x {(iy)}^{2} + {(iy)}^{3} = y + iv[/tex]

[tex]\rm \longmapsto\: {x}^{3} + 3 {x}^{2}yi + 3x {i}^{2} {y}^{2} + {i}^{3} {y}^{3} = y + iv[/tex]

We know,

[tex] \purple{\rm \longmapsto\:\boxed{\tt{ {i}^{2} = - 1 \: \: \: and \: \: \: {i}^{3} = - i }}} \\ [/tex]

So, using this, we get

[tex]\rm :\longmapsto\: {x}^{3} + 3 {x}^{2}yi - 3x{y}^{2} - i{y}^{3} = y + iv[/tex]

[tex]\rm \longmapsto\: ({x}^{3} - 3x {y}^{2}) + i(3{x}^{2}y- {y}^{3})= y + iv[/tex]

On comparing real and Imaginary parts, we get

[tex]\rm \longmapsto\:y = {x}^{3} - 3 {xy}^{2} [/tex]

and

[tex]\rm \longmapsto\:v = {3x}^{2}y - {y}^{3} [/tex]

Now, Consider

[tex]\rm \longmapsto\:\dfrac{y}{x} + \dfrac{v}{y} [/tex]

[tex]\rm \:  =  \: \dfrac{ {x}^{3} - {3xy}^{2}}{x} + \dfrac{ {3yx}^{2} - {y}^{3} }{y} [/tex]

[tex]\rm \:  =  \: \dfrac{ x({x}^{2} - {3y}^{2})}{x} + \dfrac{y({3x}^{2} - {y}^{2})}{y} [/tex]

[tex]\rm \:  =  \: {x}^{2} - {3y}^{2} + {3x}^{2} - {y}^{2} [/tex]

[tex]\rm \:  =  \: 4{x}^{2} - {4y}^{2}[/tex]

[tex]\rm \:  =  \: 4({x}^{2} - {y}^{2})[/tex]

Hence, Proved

Related Questions

Please help me oh me a lot to me and you get a lot of points I think

Answers

Answer:

A. 0.2, 0.25, 0.2

B. the scores are a quarter of the whole

Answer:

so the three numbers can be turned into fractions.

1/5, 1/4, and 4/20.

fractions can be turned into decimals by dividing the denominator into the numerator.

now we have 0.2 , 0.25 , and 0.2

greatest to least, the numbers are 0.25 , 0.2 , and 0.2

(both 0.2s are equal, so order of those doesn't matter)

Step-by-step explanation:

⅕ of shots

Dwayne 25%

4 out 20

Write each score as decimal

Compare the three score.

what is 0.25+1/8+24+60=

Answers

Answer:

84.375 or 675/8

Step-by-step explanation:

Write an equation of the line that is parallel to the y-axis and passes through the point (−3,1). Then determine whether the slope of the line is positive, negative, zero, or undefined.

Answers

Answer:

equation : x = -3

slope : undefined

Step-by-step explanation:

The equation says that it is parallel to y-axis, meaning all the x values are the same. the slope is undefined because of (HOY / VUX)

find the nth term of 11, 20, 35, 56, 83, . . .

Answers

nth number is 83 hope you understand my answer thanks

Answer: 3n^2 + 8

Step-by-step explanation:

an^2 + bn + c

It is a quadratic sequence

11, 20, 35, 56, 83

 +8 +15 +21 +27

    +6   +6   +6

a + b + c

4a + 2b + c

9a + 3b + c

4a + 2b + c - a + b + c = 20 - 11

4a + 2b + c - a + b + c = 9

3a + b = 9

9a + 3b + c - 4a + 2b + c = 35 - 20

5a + b = 15

-2a = -6

a = -6/-2

a = 3

Substitute to find the values of b & c

For b

5(3) + b = 15

15 + b = 15

b = 15 - 15

b = 0

For c

9(3) + 3(0) + c = 35

27 + c = 35

c = 35 - 27

c = 8

3n^2 + 0n + 8

3n^2 + 8

1/x^12 times x^2 please help i need to hurry with this

Answers

Answer:

x= 3, -4

Step-by-step explanation:

BIG BRAIN

The component form of a vector indicates the changes in x and y from the
starting point to the endpoint. What is the component form of this vector?

Answers

The vector’s component form is (-8, 2)

It moves negative 8 units towards the left. 4 - 8 = -4. And it moves positive 2 units upwards -1 + 2 = 1.

We will see that the component form of the given vector is <-8, 2>, so the correct option is D.

How to get the component form of the vector?

For a vector with starting point (a, b) and endpoint (c, d), the component form is:

<c -a, d -b>

In this case, we can see that the starting point is (4, -1), and the endpoint is (-4, 1)

Then the component form of the vector is:

<-4 - 4, 1 - (-1) > = <-8, 2>

So the correct option is D.

If you want to learn more about vectors, you can read:

https://brainly.com/question/3184914

28-(3x+4)=2(x+6)+x How would you solve this??? Please add step by step explanation

Answers

Answer:

x = 2

Step-by-step explanation:

Can someone please help me with number 35?

Answers

Answer:

4

Step-by-step explanation:

John Paulo bought 10 plastic discs. Each disc weighs 0.9 ounces. How much do the 10 plastic discs weigh in all?

Answers

Answer:

9

Step-by-step explanation:

.9*10=10

it's basically simple multiplication with a decimal

For w(x)=4x-4 find w(-7) show your work

Answers

Answer:

-32

Step-by-step explanation:

To find w(-7) we have to substitute x with -7

w(x)=4x-4

w(x = -7) = 4(-7) -4

w(-7) = -28-4

w(-7) = -32

-32 , I did the same question last week on my test I don’t have the work though

Please answer for last question

Answers

Step-by-step explanation:

since you know already that it is a linear equation with a constant change rate (slope), you don't know the equation itself yet ?

you found the change rate (slope).

it is x goes up by 2, and y goes down by 6.

the slope is the ratio y/x representing the y and x changes, and it is always the factor of x.

"down" or "left" means "-","up" or "right" means "+".

the slope here is therefore

-6/2 = -3/1 = -3

so, we know the line function looks like

y = -3x + b

to get b we use one of the x y pairs like (1. 4)

4 = -3×1 + b

7 = b

and the full line function is

y = -3x + 7

For BRAINILY please help I need to pass this class

Answers

This is an isosceles triangle, which means that is has two congruent legs and two congruent base angles. Therefore, because IJ and IH are congruent, angles J and H are also congruent.

v = 54 degrees

Hope this helps!! :)

May I please receive help?

Answers

Answer:

1 and 2 are parallel. 1 and 3 are perpendicular. 2 and 3 are perpendicular.

Answer: I'm gonna try my best. 1). neither 2). Parallel 3).perpendicular

Step-by-step explanation:

How do you solve this

Answers

Answer:

x = 8

Step-by-step explanation:

1. First, we should realize that the angle with 16x + 7 and the angle with 5x + 5 are same-side interior angles, meaning they add up to 180 degrees. This means that 16x + 7 + 5x + 5 = 180.

2. (Solving)

Step 1: Simplify both sides of the equation.

[tex](16x+5x)+(7+5)=180[/tex] [tex]21x + 12 = 180[/tex]

Step 2: Subtract 12 from both sides.

[tex]21x + 12 - 12 = 180 - 12[/tex] [tex]21x = 168[/tex]

Step 3: Divide both sides by 21.

[tex]\frac{21x}{21}=\frac{168}{21}[/tex] [tex]x = 8[/tex]

Step 4: Check if solution is correct.

[tex]16(8)+7+5(8)+5=180[/tex] [tex]128 + 7 + 40 + 5 = 180[/tex] [tex]135 + 40 + 5 = 180[/tex] [tex]175 + 5 = 180[/tex] [tex]80= 80[/tex]

Therefore, x = 8.

the width of a rectangular room is 3m shorter than its length . if its perimeter must not exceed 42m , calculate the greatest length the room can have​

Answers

The width of a rectangular room is 3 m shorter than its length. If its perimeter must not exceed 42 m, calculate the greatest length that the room can have.

Please help!

Suppose that the cross section of a piece of pipe looks like the diagram shown. Find the total outer diameter.

Answers

Answer:

-8

Step-by-step explanation:

A poll was taken to determine which candidate students were voting for in an upcoming class president election. Of the students who participated in the poll, 40% voted for Iverson and 50% voted for Vicky. There were 30 votes for Vicky.

There were
(blank) people who participated in the poll.

Answers

Answer:

Step-by-step explanation:

the way to find how many participanted in the class voted, you need to look at the percentages and the number of students that voted for Vicky.

50% of students voted for Vicky and the question later explained that the total of the 50% was 30 students. This mean we know half of the student that participated. To find the total amount of students to make 100%, you need to multiple 30 by 2. which is 60.

Now that we know there were 60 students in total, we can find how many participated by using Iverson's percentage.

this is the formula that can help with any percentage math problem:

The percentage of something ÷ 100% = the number you are looking for ÷ the total

in our case:

40÷100 = x (because it is the unknown term)÷60

from here you cross multiple getting

2400÷100x

which equals

24

now we have to add Vicky's and Iverson's voter

24 + 30= 54

60-54=6

6 students did not participate

Pls help
bobik: yes, or no

Answers

Answer:

yes

Step-by-step explanation:

Answer:

?? Yes??

Step-by-step explanation:

Please please help me with this question please now and show your steps

Answers

Answer:

[tex]y = {x}^{2} - 8x + 7 \\ a = 1 \: and \: b = - 8 \\ the \: line \: of \: symmetry \: \\ x = \frac{ - b}{2a} = \frac{8}{2} \\ x = 4(line \: of \: symmetry) \\ y = {4}^{2} - 8(4) + 7 \\ = 16 + 32 + 7 = y = 55 \\ vertix \: is \: (4 ,55) \\ y = {0}^{2} - 8(0) + 7 \\ y = 7 \\ 0 = {x}^{2} - 8x + 7 \\ (x - 7)(x - 1) = 0[/tex]

John makes $8.25 per hour after he got a 50 cent raise. Write the equation

Answers

Answer:

8.25+50

Step-by-step explanation:

mike wants to buy the same number of balloons for 2 of his friends. her has $59, and each balloon costs $7. What is the largest number of balloons that mike buys for each friend

Answers

Answer:

4

Step-by-step explanation:

7x8=56 wich is the highest number it can go to

so 8/2=4

What is the sum of V-2 and V-18?

Answers

Answer:

[tex] \sqrt{ - 2} + \sqrt{ - 18} \\ = \sqrt{2} i + \sqrt{ - 2} \sqrt{9} \\ = \sqrt{2} i + 3\sqrt{2} i \\ = 4\sqrt{2} i[/tex]

Jimmy wants to enlarge a sign for his taco stand. He increases the height by a scale factor of 4 and the width by a factor of 2. What is the effect on the area of the sign? times greater

Answers

After the increase in the dimensions of the sign, the effect on the area of the sign is that the new area is 8 times larger than the initial area.

The sign has the shape of a rectangle. A rectangle has four sides. The opposite sides are equal in length.

The area of a rectangle = length x width.

Jimmy is making the sign bigger by increasing the size of the length and width. Let's imagine that the width is 8cm and the length is 10cm.

The initial area of the sign = 10 x 8 = 80 cm

After the increase in the dimensions, the length is are 40cm and 16cm for the width.

New area = 40 x 16 = 640

Number of times the new area is greater than the initial area = 640 / 80 = 8 times larger

A similar question was answered  here: https://brainly.com/question/18452788

Using the quadratic formula to solve x- + 20 = 2x, what are the values of x?

Answers

Answer:

they is no answer because x-+20 is not applicable to math

Answer:

x = -20

Step-by-step explanation:

because there is no quadratic equation there

Please help me with this. Thanks!

Answers

Answer:

the plane is descending at a rate of 8,300 feet per minute

Step-by-step explanation:

31,275 - 22,975 = 9,300

BRAINLIEST PLEASE!

Solve 2x^2-3x-5=0 with the quadratic formula please and have a bless day

Answers

Answer:

( 2x^62- 3x) - 5 = 0

2x^62 - 3x - 5 = 0

X = - 1

- 5/2

The it is my dear friend

Someone plz help me :(

Answers

Answer:

i can JUST GET ON YOUR CROME BOOK

Step-by-step explanation:

Solve for x, y and z ( help ♡ )

Answers

Ans of your question . in which class do u read?

Find the distance between points (1, 3) and (9, 18) on the coordinate plane.

Answers

The correct distance is 17 via the distance formula :)
the formula is 17 via

GIVING BRAINLIEST PLS FAST

Answers

Answer: is not a full square

Step-by-step explanation:

Just look at the degree ; if it were a multiple of two , then the number would be a full square ; but as we see , 9 is not a multiple of two  , from which it is not a full square

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