If (x+iy)³ = y+vi then show that (y/x + v/y) = 4(x²-y²)
Answers
Answer:
Step-by-step explanation:
[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
If you receive $68 for mowing lawns for 8 hours. What would be the equation to find out how much money you would receive in an hour?
Answers
Answer:
$68÷8= x money per hour
Step-by-step explanation:
explain and show how please ! and give answer someone pls answer asap !
Answers
Answer:
Both angels are 70 and both added are 140.
Step-by-step explanation:
Because of the rules, that's names i never remember angle 1 and angle 2 are both 70° and obiously they equal 140.
3. Which table has a constant of proportionality equal to 2.5?
x
х
y
X
0
0
2.5
os
2.5
NO
1
2
.
5
7.5
А
о
B
х
y
2
у
2.5
3.75
W NX
2
5
3
3
7.5
10
4
ch
5
4
D
Answers
Answer:
your answer is A
I can assure you that you will give brainlest
Answer:
A
Step-by-step explanation:
[tex]The \: value \: of \: \sqrt{6 + \sqrt{6 + \sqrt{6} + .......... \: to \: \infin } } \: is \: \underline{ \: \: \: \: \: \: \: \: \: \: \: .}[/tex]
CORRECT EXPLANATION WILL BE MARK AS BRAINLIEST.
Answers
[tex]\large\underline{\sf{Solution-}}[/tex]
Let us assume that:
[tex] \sf \longmapsto x = \sqrt{6 + \sqrt{6 + \sqrt{6 + ... \infty } } } [/tex]
We can also write it as:
[tex] \sf \longmapsto x = \sqrt{6 + x } [/tex]
Squaring both sides, we get:
[tex] \sf \longmapsto {x}^{2} =6 + x[/tex]
[tex] \sf \longmapsto {x}^{2} - x - 6 =0[/tex]
By splitting the middle term:
[tex] \sf \longmapsto {x}^{2} - 3x + 2x - 6 =0[/tex]
[tex] \sf \longmapsto x(x - 3) + 2(x - 3 ) =0[/tex]
[tex] \sf \longmapsto (x+ 2)(x - 3 ) =0[/tex]
Therefore:
[tex] \longmapsto\begin{cases} \sf (x+ 2) =0 \\ \sf (x - 3) = 0 \end{cases}[/tex]
[tex] \sf \longmapsto x = - 2,3[/tex]
But x cannot be negative.
[tex] \sf \longmapsto x = 3[/tex]
Therefore, the value of the expression is 3.
[tex]\large\underline{\sf{Verification-}}[/tex]
Given:
[tex]\sf\longmapsto x=3[/tex]
We can also write it as:
[tex]\sf\longmapsto x = \sqrt{9}[/tex]
[tex]\sf\longmapsto x = \sqrt{6+3}[/tex]
[tex]\sf\longmapsto x = \sqrt{6 + \sqrt{9}}[/tex]
[tex]\sf\longmapsto x = \sqrt{6 + \sqrt{6+3}}[/tex]
[tex]\sf\longmapsto x = \sqrt{6 + \sqrt{6+\sqrt{9}}}[/tex]
[tex]\sf\longmapsto x = \sqrt{6 + \sqrt{6+\sqrt{6+3}}}[/tex]
This pattern will continue.
[tex]\sf\longmapsto x = \sqrt{6 + \sqrt{6+\sqrt{6+...\infty}}}[/tex]
Solve for x 3/8(2x+16)−2=13
The "/" is for a fraction btw
Answers
I hope this helped
ANSWER THIS ASAP PLEASE !
Answers
Simple the answer is: n=14
We can move the divided by to other side of equal sign and it now because multiply by 5.
2n=28
We add 3 to both sides of equal sign
n=14
We divide 2 to both sides of equal sign to get the variable by itself
Your answer is : n=14
A student performed the following steps to find the solution to the equation
x2 - 2x - 8 = 0. Where did the student go wrong?
Step 1. Factor the polynomial into (x-4) and (x - 2)
Step 2. X-4 = 0 and x - 2 = 0
Step 3. x = 4 and x = 2
A. The student did not make any mistakes, the solution is correct
B. in Step 2
C. in Step 1
D. in Step 3
Answers
Answer:
step 1, c
Step-by-step explanation:
he was wrong in step 1.
when factoring out (x-4) and (x-2), you have to distribute x*x, x*-2, -4*x, and -4*-2
if you do it you get x^2 - 2x -4x + 8
the correct way to factor will be (x-4)(x+2)
when you factor out you will get the original equation you started with.
b(n)=−4−2(n−1)
find the 12th term in the sequence.
Answers
Answer:
-18
Step-by-step explanation:
b(12) = 4 - 2(12-1)
= 4 - 2(11)
= 4 - 22
= -18
will give BRAINLIEST!!!!!!
3 3/4 X 1 2/5 do a step by step explanation please and also you have to make it so you convert it into an improper fraction first... please answer!!!
Answers
Answer:
5 1/4
Step-by-step explanation:
To convert into an improper fraction, we times the whole number by the denominator and then add the numerator. For the first number, it is 15/4 and the second number is 7/5. Then we multiply 15/4 and 7/5 to get 5 1/4.
Answer:
99/5
Step-by-step explanation:
CONVERT INTO:
33/4 x 12/5
MULTIPLY THE FRACTIONS:
(33x12)
--------------
5
CANCEL TERMS:
(33x3)
-----------
5
SIMPLIFY:
99/5
please help I will make you Brainliest if correct (show working)
Answers
Split the irregular shape into 2 (A and B)
A= 4m x 3.8m = 15.2m^2
B= 3m x 2.8m = 8.4m^2
2.) Total area = A + B
= 15.2 + 8.4 = 23.6m^2
if 3x - 2y = 9 , 27x³- 8y³= 1377 , prove that xy = 4
plzz answer this question...
Answers
Step-by-step explanation:
Here we are given that , the Value of
[tex] 3x - 2y = 9 [/tex] [tex] 27x^3 - 8y^3 = 1377 [/tex]And we need to find the prove that ,
[tex] xy = 4 [/tex]We know a identity as ,
[tex] (a-b)^3 = a^3-b^3-3ab(a-b) [/tex]On cubing both sides of the given equation,
[tex] (3x-2y)^3 = 9^3 [/tex]Simplify and then substitute ,
[tex] 27x^3-8y^3 -3(3x)(2y)(3x-2y ) = 729[/tex] [tex] 1377 - 18xy (9) = 729 [/tex][tex] 162xy = 1377-729 [/tex][tex] xy =\dfrac{648}{162}[/tex][tex]xy = 4 [/tex]Hence Proved !
What is the value of {-12}
–13
–12
12
13
Answers
Answer: B. -12
Edge 21: Since the brackets are facing down, there is no need to round the -12 because it's already an integer.
Is it possible to list all the fractions between 2 and 3? Explain.
Answers
Answer: No
Step-by-step explanation:
There are infinitely many numbers. Numbers so big that they are beyond our comprehnsion. Numbers can also be infinitly small. A fraction is something less than a whole. So the answer is no you can't list them all.
maths help me maths giving 10 point
Answers
Step-by-step explanation:
[tex]x-2>4[/tex]
[tex]x>6[/tex]
And
[tex]3x+1\leq 28[/tex]
[tex]3x\leq 27[/tex]
[tex]x\leq 9[/tex]
So [tex]x[/tex] ∈ [tex](6, 9][/tex]
Please give me a brainliest answer
3(x+1)=2(x-3) whats the answer
Answers
The answer is -9
I dont know the step-by-step used a calculator
Can someone help me and if you don’t have the answer don’t answer cause you’re just wasting my time
Answers
Answer: 1 is 534
Step-by-step explanation:
Help me please i need the equation of axis of symmetry
Answers
Answer:
x = -2
Step-by-step explanation:
It's a vertical line at x = -2
Use synthetic division to find the result when 4x^3+3x^2– 8x – 7 is divided by x+1.
Answers
4x^2 - x - 7
Jerry played with 3 friends today, all at different times. He played with Tommy for 1 7/8 hours, then Nathan for 1 3/4 hours, then Chris for 3/2 hours.
How many hours did Jerry play with his 3 friends today?
Answers
Answer:
4hrs 7 minutes and 30 seconds
Step-by-step explanation:
1 7/8 = (1=60) + (7/8 * 60) = 1 : 52:50 time played with Tommy 1 3/4 = (1=60 ) ( (3/4 * 60) = 1:45:00 time played with Nathan 3/2 = (3/2 *60) = 90 = 1:30:00 time played with Chris Total played with 3 friends = 1:52:50 + 1:45:00+1:30:00= 4:07:50 = 4hrs 7 minutes and 30 seconds
PLEASE HELPP!! Type an equation for the
following pattern. Make sure
the fraction is in simplest form.
Answers
Answer:
y = -3/2x + 10.5
Step-by-step explanation:
You can plug in the x values and you'll get the y values of the table. Hope this helps :)
Which pair of numbers has a sum of 0?
1) 2 and 12
2) 2 and –12
3) 2 and –0.2
4) 2 and –2
Answers
If you did 2 + - 2, it is equal to 2 - 2 which is 0
Apply the distributive property:
-3(x+2) =
Answers
Answer:
-3x-6
Step-by-step explanation:
you multiply x by -3 getting -3x then you multiply -3 by positve 2 getting -6
Hello skm413002!
[tex] \huge \boxed{\mathfrak{Question} \downarrow}[/tex]
Apply the distributive property:
-3(x+2) =
[tex] \large \boxed{\mathbb{ANSWER\: WITH\: EXPLANATION} \downarrow}[/tex]
By applying the distributive property, we can solve it as....
[tex] \sf \: - 3(x + 2) \\ = \sf \: (- 3 \times x) -( 3 \times 2) \\ = \large \boxed{\boxed{ \bf - 3x - 6}}[/tex]
Method :-Distributive property.Steps :-Multiply -3 with x & 2.Add the 2 products.You'll get the answer as -3x - 6.__________________
Hope it'll help you!
ℓu¢αzz ッ
5.5.WW3
Question Help
Find the quotient
30)593
Select the correct choice below and fill in the answer box(es) within your choice.
OA. The quotient is
OB. The quotient is
R
Answers
Because it just makes since I think
Write the prime factorization of 42. Use exponents when appropriate and order the factors from least to greatest
Answers
2 x 3 x 7 = 42
Answer:
42 = 2*3*7
Step-by-step explanation:
42 can be rewritten as 2*21, for example.
2 is prime. 21 = 3*7.
Then 42 = 2 * 21 is also equal to:
42 = 2*3*7. where these factors are ordered from least to greatest. The prime factors of 42 are {2, 3, 7}.
what is the sign of a • (-b/b) when a = 0 and b < 0?
Answers
Answer:
gimmie my points boi
Step-by-step explanation:
eiurfh4iewurhfwo8rho
Can you translate the measurements to (cups, teaspoons, tablespoons etc.)
Top flour 170g
Baking powder 5g
Sugar 10g
Butter 40g
Maple syrup 30ml
Milk 40ml
Answers
Answer:
Flour: 18.76 Tablespoons
56.28 Teaspoons
1.17 U.S. Cups
Baking Powder: 0.35 Tablespoons
1.04 Teaspoons
0.02 U.S. Cups
Sugar: 0.80 Tablespoons
2.40 Teaspoons
0.05 U.S. Cups
Butter: 2.82 Tablespoons
8.46 Teaspoons
0.18 U.S. Cups
Maple Syrup:
0.127 Cups
2.029 Tablespoons
6.087 Teaspoons
Milk:
0.169 Cups
2.705 Tablespoons
8.115 Teaspoons
Hope this helps!!
Have A Blessed Day <333
Solve for X then find the measure of
Answers
Answer:
x=11
m<a=80º
m<b=80º
Step-by-step explanation:
They are congruent so
8x-8=5x+25
8x=5x+33
3x=33
x=11
Answer:
Step-by-step explanation:
8x-8=5x+25
8x+5x=25--8
13x=33
x=2.5
x=3cm
Select the correct answer.
Which statement is true about the graph of this equation?
y + 4 = 4(x + 1)
A.
The graph is a line that goes through the points (1,-4) and (0,0).
B.
The graph is a line that goes through the points (1,4) and (2,8).
C.
The graph is a line that goes through the points (-4,-1) and (-3,-3).
D.
The graph is a line that goes through the points (4,1) and (5,5).
Answers
I hope this helped you !!
Answer:
A
Step-by-step explanation:
The graph is a line that goes through the points (1,-4) and (0,0)
if i had 475 cannolis ans a box can only carry 15 how many boxes will i need.
Answers
Answer:
you would need 32 boxes
Step-by-step explanation:
475/15 = 31.666
32 boxes
475/15=31.7 but you can't get 0.7 boxes so you would need 32 to carry all of the cannolis.
Can someone help me with these two? Please
Answers
For 10/15 it would be 2/3
Answer:
8/9 and 2/3
Step-by-step explanation:
find common factor with each of the numbers and divide by that
16 and 18, 2 is the gcf.
16/2=8 18/2=9
the gcf in the second one is 5.
10/5=2 15/5=3
Hope this helps!