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
steposvetlana [31]
3 years ago
14

Find the real numbers x and y if -3+ix^2y and x^2+y+4i are conjugate of each other. Pls solve with the steps

Mathematics
2 answers:
nasty-shy [4]3 years ago
6 0

-3+ix^2y\\x^2+y+4i\\\\x^2+y=-3\\ix^2y=-4i\\\\x^2=-3-y\\x^2=-\dfrac{4}{y}\\\\-3-y=-\dfrac{4}{y}\\3y+y^2=4\\y^2+3y-4=0\\y^2-y+4y-4=0\\y(y-1)+4(y-1)=0\\(y+4)(y-1)=0\\y=-4 \vee y=1\\\\x^2=1 \vee x^2=-4\\x=-1 \vee x=1\\\\\boxed{(x,y)\in\{(-1,-4),(1,-4)\}}

Firdavs [7]3 years ago
4 0
ANSWER
x = ±1 and y = -4.
Either x = +1 or x = -1 will work

EXPLANATION
If -3 + ix²y and x² + y + 4i are complex conjugates, then one of them can be written in the form a + bi and the other in the form a - bi. In other words, between conjugates, the imaginary parts are same in absolute value but different in sign (b and -b). The real parts are the same

For -3 + ix²y
⇒ real part: -3
⇒ imaginary part: x²y

For x² + y + 4i
⇒ real part: x² + y (since x, y are real numbers)
⇒ imaginary part: 4

Therefore, for the two expressions to be conjugates, we must satisfy the two conditions. 

Condition 1: Imaginary parts are same in absolute value but different in sign. We can set the imaginary part of -3 + ix²y to be the negative imaginary part of x² + y + 4i so that the 

   x²y = -4 ... (I)

Condition 2: Real parts are the same

   x² + y = -3 ... (II)

We have a system of equations since both conditions must be satisfied

   x²y = -4 ... (I)
   x² + y = -3 ... (II)

We can rearrange equation (II) so that we have

   y = -3 - x² ... (II)

Substituting into equation (I)

   x²y = -4 ... (I)
   x²(-3 - x²) = -4
   -3x² - x⁴ = -4
   x⁴ + 3x² - 4 = 0
   (x² + 4)(x² - 1) = 0
   (x² + 4)(x-1)(x+1) = 0

Therefore, x = ±1.
Leave alone (x² + 4) as it gives no real solutions.

Solve for y:

   y = -3 - x² ... (II)
   y = -3 - (±1)²
   y = -3 - 1
   y = -4

So x = ±1 and y = -4. We can confirm this results in conjugates by substituting into the expressions:

   -3 + ix²y 
   = -3 + i(±1)²(-4)
   = -3 - 4i

   x² + y + 4i
   = (±1)² - 4 + 4i
   = 1 - 4 + 4i
   = -3 + 4i

They result in conjugates
You might be interested in
A number w increased by 6 is at least 24. Using an inequality
Mekhanik [1.2K]

Answer:

x<24

Step-by-step explanation:

4 0
2 years ago
Read 2 more answers
Helpppppppppppppppppp!!!!
Temka [501]

Answer:

Don't worry Its a

Step-by-step explanation:

3 0
3 years ago
Read 2 more answers
What is the m∠ACB in the circle shown below?
GuDViN [60]

Question:

What is the m∠ACB in the circle shown below?

Answer:

45

Step-by-step explanation:

So we know it is <u><em>less than a right angle</em></u> and it is not a obtuse angle it is less than 90 degrees. 45 is less, 90 is equal(<em>not what we are looking for</em>) , 120, is greater, and 180 is greater which leaves us to our answer 45.

4 0
2 years ago
I need help with this whats the answers on #1 i suck at geometry plz
sammy [17]
Z is congruent to q.
4 0
3 years ago
Please Help! Easy!
irinina [24]

Answer:

The answer is B.

4 0
3 years ago
Read 2 more answers
Other questions:
  • What is the quotient?
    5·1 answer
  • Bill wants to rent a car. Rental Company A charges $35 per day plus $0.10 per mile driven. Rental Company B charges $25 per day
    11·1 answer
  • Tell whether (2,6) is a solution of y = 3x. (show your work)
    7·1 answer
  • What is the slope of the line that contains these points x=12,13,14,15 y=-4,2,8,14
    5·1 answer
  • ________ took the law into their own hands and acted as police, judge, jury and sometimes executioner​
    7·1 answer
  • PLEASE HELP ITS ATEST I BEG U. NEED HELP
    13·2 answers
  • Idenify patterns using a ?
    7·1 answer
  • What is 89999999+ 3099999999999=??????
    8·2 answers
  • The third graders at Grace Hardeman Elementary want to play kickball. There are 20 students, and each team will have 5 students.
    12·1 answer
  • William and Shana plan to stain the concrete surface of their square-shaped driveway and of their patio.
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!