(a) (i) Calculate (4 + 10i)². (ii) Hence, and without using a calculator, determine all solutions of the quadratic equation z² +8iz +5-20i = 0. (b) Determine all solutions of z² +8z +7= 0.
Answers
(a) The solutions of the quadratic equation are -4i + 2√11i and -4i - 2√11i and (b) the solutions of the quadratic equation are -1 and -7.
(a) (i) To calculate (4 + 10i)², we'll have to expand the given expression as shown below:
(4 + 10i)²= (4 + 10i)(4 + 10i)= 16 + 40i + 40i + 100i²= 16 + 80i - 100= -84 + 80i
Therefore, (4 + 10i)² = -84 + 80i.
(ii) We are given the quadratic equation z² + 8iz + 5 - 20i = 0.
The coefficients a, b, and c of the quadratic equation are as follows: a = 1b = 8ic = 5 - 20i
To solve this quadratic equation, we'll use the quadratic formula which is as follows:
x = [-b ± √(b² - 4ac)]/2a
Substitute the values of a, b, and c in the above formula and simplify:
x = [-8i ± √((8i)² - 4(1)(5-20i))]/2(1)= [-8i ± √(64i² + 80)]/2= [-8i ± √(-256 + 80)]/2= [-8i ± √(-176)]/2= [-8i ± 4√11 i]/2= -4i ± 2√11i
Therefore, the solutions of the quadratic equation are -4i + 2√11i and -4i - 2√11i.
(b) We are given the quadratic equation z² + 8z + 7 = 0.
The coefficients a, b, and c of the quadratic equation are as follows: a = 1b = 8c = 7
To solve this quadratic equation, we'll use the quadratic formula which is as follows: x = [-b ± √(b² - 4ac)]/2a
Substitute the values of a, b, and c in the above formula and simplify:
x = [-8 ± √(8² - 4(1)(7))]/2= [-8 ± √(64 - 28)]/2= [-8 ± √36]/2= [-8 ± 6]/2=-1 or -7
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Related Questions
in a class of 23 students, 7 have brothers and 5 have sisters. There are 2 students who have a brother and a sister. what is the probability that a student who has a brother also has a sister?
Answers
The probability that a student who has a brother also has a sister is 2/7 or approximately 0.2857.
To find the probability that a student who has a brother also has a sister, we need to use the concept of conditional probability.
Let's first find the probability of a student having both a brother and a sister. According to the problem, there are 2 students who have both a brother and a sister. Therefore, the probability of a student having both a brother and a sister is:
P(Brother and Sister) = 2/23
Now, let's find the probability of a student having a brother:
P(Brother) = 7/23
Using the formula for conditional probability, we can find the probability that a student who has a brother also has a sister:
P(Sister|Brother) = P(Brother and Sister) / P(Brother)
Substituting the values we found, we get:
P(Sister|Brother) = (2/23) / (7/23) = 2/7
This means that if we randomly select a student who has a brother, the probability that they also have a sister is about 0.2857.
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the boxplot below represents annual salaries of attorneys in thousands of dollars in los angeles. about what percentage of the attorneys have salaries between and ? a. b. c. d. e. none of the above
Answers
The percentage of attorneys in given range is approximately 50%.
What is percentage of the attorneys?The boxplot below represents the salaries of attorneys in Los Angeles.
We can estimate the percentage of attorneys who have salaries between $55,000 and $80,000 by counting the number of boxes between the range (called the interquartile range, or IQR), which is from $55,000 to $80,000.
There are two boxes within the range, and two boxes that are not within the range (called the lower and upper outliers).
Thus, the percentage of attorneys with salaries between $55,000 and $80,000 is approximately 50%.
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WILL GIVE BRAINLIEST The number of milligrams of a certain drug that is in a patient's bloodstream h hours after the drug is injected is given by the following function.
When the number of milligrams reaches 6 , the drug is to be injected again. How much time is needed between injections?
Round your answer to the nearest tenth, and do not round any intermediate computations.
Answers
Answer:
3.6 hours
Step-by-step explanation:
Given that the number of milligrams of a drug in a patient's bloodstream after h hours is D(h) = 25e^(-0.4h), you want to know how much time elapses before the number reaches 6 milligrams.
SolutionYou want to solve for h the equation ...
D(h) = 6
25e^(-0.4h) = 6 . . . . . . use the given expression for D(h)
e^(-0.4h) = 6/25 . . . . . divide by 25
-0.4h = ln(6/25) . . . . . take natural logs
h = ln(6/25)/-0.4 ≈ 3.6
About 3.6 hours is the time needed between injections.
__
Additional comment
Because there are 6 mg remaining after the first injection when the second is given, it will take about 4.1 hours for the amount to decline to 6 mg after the second injection.
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What steps should be taken to calculate the volume of the prism using the formula V = Bh?
A triangular prism. The rectangular sides are 5 meters by 3.75 meters. The triangular sides have a base of 3.75 meters and height of 3 meters.
Check all that apply.
Use the triangle base and height to calculate the area of the base, B.
The area of the base, B, is One-half(3)(5) = 7.5 m2.
The area of the base, B, is One-half(3.75)(3) = 5.625 m2.
Substitute the values in for the variables and simplify.
Label the answer with the correct units.
Answers
In this scenario, the steps that should be taken to calculate the volume of a prism are:
⇒ A, C, D, and E.
Now, Mathematically, the volume of a prism should be calculated by using this formula:
⇒ V = B × H
Where:
H is the height.
B is the base area.
Hence, In conclusion, you should use the triangle-base and height to solve for the area of the base, and the area of the base (B) is one-half of the product of (3.75) and (3), which is,
= 1/2 x 3.75 x 3
= 5.625 m².
Thus, the steps that should be taken to calculate the volume of a prism are:
⇒ A, C, D, and E.
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The pH scale measures how acidic or basic a substance is. Lemon juice is said to have a pH of less than 4 and greater than 1.5. Model the normal range of pH values of lemon juice, using a compound inequality.
Answers
Answer:
1.5<x<4 i hope this is right sorry if its not
Step-by-step explanation:
Answer:
b
Step-by-step explanation:
3(-6x -5) = 75 pls help me
Answers
Answer:
x = -5
Step-by-step explanation:
3(-6x -5) = 75
Divide each side by 3
3/3(-6x -5) = 75/3
-6x-5 = 25
Add 5 to each side
-6x-5+5 = 25+5
-6x = 30
Divide each side by -6
-6x/-6 = 30/-6
x = -5
505
Autumn has 6 ribbons for her hair. Each ribbon is 3 feet long. How many total inches of ribbon
does Autumn have?
Answers
Answer:
she has a total of 216 inches of ribbon
Step-by-step explanation:
1 foot is equal to 12 inches.
if she has 3 feet per ribbon, and each foot is 12 inches, then 3 × 12 would be 36, making each ribbon 36 inches.
But, she has 6 pieces of ribbon, so that's 36 inches 6 times. 6 × 36 is 216, so she has a total of 216 inches of ribbon.
last month 15 homes were sold in town x. the average (arithmetic mean) sale price of the homes was $150,000 and the median sale price was $130,000. which of the following statements must be true? i. at least one of the homes was sold for more than $165,000. ii. at least one of the homes was sold for more than $130,000 and less than $150,000. iii. at least one of the homes was sold for less than $130,000.
Answers
Statement ii "at least one of the homes was sold for more than $130,000 and less than $150,000." must be true. Because Since the arithmetic mean sale price is $130,000, it means that half of the homes were sold for more than $130,000 and half were sold for less than $130,000. So, the correct option is Statement ii.
Since the arithmetic mean sale price is $150,000, it means that the total sale price of all 15 homes combined was $150,000 x 15 = $2,250,000. However, this does not necessarily mean that at least one of the homes was sold for more than $165,000, as some homes could have been sold for less than the mean to bring the average down. Therefore, statement i is not necessarily true.
On the other hand, statement iii is not necessarily true either, as we have no information about the sale price of the lowest-priced home or any other individual home. It is possible that all 15 homes were sold for more than $130,000, in which case statement iii would be false.
So, the correct answer is statement ii.
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Which statement describes the mapping? A. The mapping represents y as a function of x, because each y-value corresponds to exactly one x-value. B. The mapping does not represent y as a function of x, because two of the x-values correspond to the same y-value. C. The mapping represents y as a function of x, because each x-value corresponds to exactly one y-value. D. The mapping does not represent y as a function of x, because there are more x-values than different corresponding y-values.
Answers
Answer:
The answer is
C. The mapping represents y as a function of x, because each x-value corresponds to exactly one y-value.
Step-by-step explanation:
In mathematics, mapping is used synonymously to a function, and a function is basically a relation between two values or expressions say y and x
like the difference equation relating y and x.
In a function, for every value of x plugged into the function there is always a corresponding value of y.
Example.
Given the function
y=2x+5.
find the value of y when x=0
we are going to substitute x=0 and solve
y=2(0)+5
y=5
so for x=0 y=5
use the alternating series test, if applicable, to determine the convergence or divergence of the series. [infinity] n = 7 (−1)nn n − 6
Answers
To apply the Alternating Series Test, we need to check two conditions:
The terms of the series must alternate in sign.
The absolute values of the terms must decrease as n increases.
Let's analyze the given series: ∑ (-1)^n (n - 6) from n = 7 to infinity.
Alternating Signs: The series has alternating signs because of the (-1)^n term. When n is even, (-1)^n becomes positive, and when n is odd, (-1)^n becomes negative.
Decreasing Absolute Values: Let's examine the absolute values of the terms: |(-1)^n (n - 6)| = |n - 6|.
As n increases, the absolute value |n - 6| also increases. Therefore, the absolute values of the terms do not decrease.
Since the terms do not meet the decreasing absolute values condition, we cannot conclude convergence or divergence using the Alternating Series Test. The Alternating Series Test does not apply in this case.
To determine the convergence or divergence of the series, we need to use other convergence tests, such as the Ratio Test or the Comparison Test.
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keith drove 320 miles using 12 gallons of gas. at this rate, how many gallons of gas would he need to drive 296 miles?
Answers
Answer:
9.87 Gallons
Step-by-step explanation:
Start by finding how far mr. keith here can drive with one gallon of gas.
to do this you will:
360 ÷ 12 = 30
For one gallon of gas, he can drive 30 miles.
Now that we know the rate for 1 gallon of gas, you just apply that rate to the new mileage:
296 (new distance) ÷ 30 (how many he can do in one mile) = 9.87 (how many times that 30 miles goes into 296 (recall that this is the rate))
To drive 296 miles, keith will use 9.87 gallons of gasoline.
hope this helped :)
Which one is different?
Answers
Step-by-step explanation: it is different because those are the only coins without a face
for a store's anniversary sale, 280 employees were asked their favorite color choice for sales tags. the results are shown in the circle graph. how many employees chose purple?
Answers
56 employees chose purple as their favorite color choice for sales tags.
The circle graph is a visual representation of the data, where each sector's angle is proportional to the percentage of the total number of employees who chose that color. This graph is an effective way to communicate complex data quickly and easily.
From the circle graph, we can see that the sector for purple covers 20% of the total circle, which represents the percentage of employees who chose purple as their favorite color choice for sales tags. To find the actual number of employees who chose purple, we need to calculate 20% of the total number of employees surveyed.
20% of 280 is equal to (20/100) x 280 = 56.
Therefore, 56 employees chose purple as their favorite color choice for sales tags.
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2. Is the following inequality always, sometimes, or never true?
2(3x - 1) +4> 6x + 2
Answers
The two sides of an inequality are not the same so inequality can never be true.
What is inequality?When two expressions are connected by a sign like "not equal to," "greater than," or "less than," it is said to be inequitable. The inequality shows the greater than and less than relationship between variables and the numbers.
Given that the inequality is 2(3x - 1) +4> 6x + 2. The inequality will be solved as below;-
2(3x - 1) +4> 6x + 2
6x - 2 + 4 > 6x + 2
6x + 2 > 6x + 2
The two sides are equal so inequality can never be true.
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7. A small high school designed a poll for parents to determine the types of field trips their families could support. There are four options on the ballot. Each family has been instructed to rate each option in order of most preferred to least preferred. If the school uses the Borda count method, how many points should be assigned to each first-place selection? 1 2 3 4
Answers
The Borda count method assigns 4 points to each first-place selection.
The Borda count method is a voting system that assigns points to each option based on their ranking in the preferences of the voters. In this case, since there are four options on the ballot and each family is instructed to rank them from most preferred to least preferred, the first-place selection should receive the highest number of points.
Since there are four options, the highest rank (first place) should be assigned 4 points. The second-place selection would receive 3 points, the third-place selection would receive 2 points, and the fourth-place selection would receive 1 point. This ensures that the most preferred option receives the highest score.
Therefore, in the Borda count method, each first-place selection should be assigned 4 points.
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solve 6x + 3 < x < 3x + 9 for integers values of x
Answers
Answer:
-9/2 < x < -3/5
Step-by-step explanation:
So first, we take the left part of the inequality to solve and ignore the right part for now, which leaves us 6x + 3 < x.
1. Subtract 3 from both sides:
6x < x - 3
2. subtract x from both sides:
5x < -3
3. divide 5 from both sides:
x < -3/5
Then, we do the same thing as part 1, but this time with x < 3x + 9.
1. Subtract 9 from both sides:
x - 9 < 3x
2. subtract x from both sides:
-9 < 2x
3. divide both sides by 2:
-9/2 < x.
Notice how we have a value that x is less than and a value that x is greater than. So now, all we have to do is to put the two inequalities together, leaving only one x in the middle. Hence, -9/2 < x < -3/5.
I hope this helped! :D
\(\\ \sf\longmapsto 6x + 3 < x < 3x + 9 \\ \\ \sf\longmapsto 6x - 3x < x < 9 - 3 \\ \\ \sf\longmapsto 3x < x < 6 \\ \\ \sf\longmapsto x < \frac{x}{3} < 2\)
a box plot is a graphical representation of data that is based on a histogram. z-scores. a five number summary. the empirical rule.
Answers
A box plot is a graphical representation of data that is based on (c) a five number summary
How to complete the statementFrom the question, we have the following parameters that can be used in our computation:
Statement: a box plot is a graphical representation of data that is based on__
A box plot (also called a box-and-whisker plot) is a way to display the distribution of a dataset.
It shows the median, quartiles, minimum, and maximum values of the data.
These items are referred to as the five number summary
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The number (√5+2)/(√5-2)
is:
(a) a rational number
(b) an irrational number
(c) an integer
(d) a natural number
Answers
Answer:
b: irrational
Step-by-step explanation:
(√5+2)
-------------- is given.
(√5 - 2)
If we multiply numerator and denominator by (√5+2), we succeed in removing the square root symbol from the denominator:
(√5+2)(√5+2)
---------------------
(√5+2)(√5- 2)
and the denominator becomes 5 - 2, or 3. The denominator is now rational, but the numerator still involves the square root symbol, so overall the given number is irrational.
The square of a certain negative number is equal to five more than one-half of that number. What is the number?
-2
-5/2
-1/2
Answers
Answer:
-2
Step-by-step explanation:
Which sequence is geometric?
•1, 5, 9, 13
•2, 6, 8, 10
•5, 7, 9, 11
•4, 8, 16, 32
Answers
Answer:
1,
Step-by-step explanation:
now z also a dk4k ak4 smka 4ml 4 LL
Economic growth typically results in rising standards of living and prosperity. However, it also invites negative externalities such as environmental degradation due to over- exploiting of natural resources. As such, the world is confronted with the dilemma of growth versus environmental sustainability. Developing a model explaining the disparity of economic development concentrating on drivers such as tourism sustainability, technological innovation and the quality of leadership would be important not only to facilitate future economic growth in developing countries, but also to the environmental and sociocultural sustainability which ultimately lead to global sustainable development. The present research objective is to develop and test framework of sustainable development by considering the elements of tourism, technological innovation, and national leadership. This further would facilitate growth, environmental and socio-cultural sustainability. Understanding the integration of these dimensions would enable the building of a Sustainable Development Framework (SDF) that would provide better insight in promoting the SDGS agenda. Ultimately, growth and environmental sustainability can be achieved which will benefit the society, the economy, and nations and of course for future sustainable policy recommendation. Based on the issue above, you are required to propose relevant econometric approaches with the aims to test sustainable development by considering the elements of tourism, technological innovation, and national leadership. Question 1 [10 marks] [CLO2] Based on the scenario above, a. Propose an appropriate model specification based on the scenario above. [4 marks] used in the [4 marks] [2 marks] b. Justify the selection of the dependent and independent variables model. c. Justify the selection of the sample period.
Answers
According to the given information, the sample period should be from 2010-2020.
a) Model specification
The model specification based on the scenario above is as follows:
SDF= f(T, TI, NL)
Where: SDF= Sustainable Development Framework
T= Tourism
TI= Technological innovation
NL= National leadership
b) Justification for the selection of the dependent and independent variables model:
Dependent variable: The dependent variable in this model is Sustainable Development Framework (SDF). The model seeks to develop a framework for sustainable development that would facilitate growth, environmental and socio-cultural sustainability.
Independent variables:
The independent variables are tourism sustainability, technological innovation, and quality of leadership. These variables drive economic development. The inclusion of tourism sustainability reflects its importance in the global economy and its potential to drive growth.
The inclusion of technological innovation reflects its potential to enhance productivity and create new industries. The inclusion of national leadership reflects the role of governance in promoting sustainable development and managing negative externalities.
c) Justification for the selection of the sample period:
The sample period should be selected based on the availability of data for the variables of interest. Ideally, the period should be long enough to capture trends and patterns in the data. However, it should not be too long that the data becomes obsolete or no longer relevant.
Additionally, the period should also reflect the context and relevance of the research question. Therefore, the sample period for this study should cover the last decade to capture the trends and patterns in the data and reflect the relevance of the research question.
The sample period should be from 2010-2020.
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Given the table and rate of 2ft per second, predict how far the car traveled in 9 seconds. Explain how you made your prediction.
Answers
Answer:
18ft
Step-by-step explanation:
if the car moves 2 ft per second, and you are trying to find how many feet the car traveled in 9 seconds. so you would have to do 2x9 to get your answer of 18. hope this helps!
Which was a challenge for the earliest permanent settlements?
Flooding allowed villagers to expand their property.
Fire risk increased due to the closeness of buildings.
Disease was a threat because of the close proximity to the rivers.
Droughts were a challenge that villagers overcame with technology..
Answers
Answer:
B
Step-by-step explanation:
Answer:
b
Step-by-step explanation:
Help me please thank u for helping me
Answers
Answer:
d
Step-by-step explanation:
Answer:
b
Step-by-step explanation:
x=1/4
1) The change in temperature from Wednesday to Thursday was -10°F. The
change in temperature from Thursday to Friday was -7°F. Which represents
a greater change in temperature?
The change from Wednesday to Thursday represents a bigger change in temperature.
The change from Thursday to Friday represents a bigger
change in temperature.
They are the same.
Answers
What I do understand of the quesetion, it should be "The change from Thursday to Friday represents a bigger change in temperature."
What is the perimeter of the triangle?
Answers
12 x 10 = 120 and 120/2 = 60 units
60 units is the answer
Sana is playing basketball.After she took 20 shots,she got in 55% of those shots.After she took 5 shots,the overall percentage of her shots that went in became 56%.How many of her last 5 shots went in ?
Answers
Answer:
3
Step-by-step explanation:
55% of 20 is 11 so she originally got 11 shots in. 56% of 20 + 5 = 25 is 14 so after the 5 extra shots she got 14 in. The answer is 14 - 11 = 3.
Answer:
hungry as can be sorry needed the points
Today, a population of bees contains 58 bees. The population grows exponentially. After 3 weeks, there will be 196 bees. How many bees will there be 10 weeks from now? Round to the nearest whole number.
Answers
Answer:
170 is what i got
Step-by-step explanation:
if you multiply 58 times 3 then you will get 171 but as a whole number you get 170
One person takes up about 2.5 square feet of space. Use this value to estimate how many people can fit
in an area that is 20 feet wide and 25 feet long.
Answers
An area that is 20 feet wide and 25 feet long will be 20*25 = 500 square feet large. Division tells us how many times one number is contained within a number, so to find how many 2.5 square feet large people fit in a 500 square feet large space, we do 500/2.5 = 200 people.
as in example 2.38, assume that 90% of the coins in circulation are fair, and the remaining 10% are biased coins that give tails with probability 3/5. i hold a randomly chosen coin and begin to flip it. (a) after one flip that results in tails, what is the probability that the coin i hold is a biased coin? after two flips that both give tails? after n flips that all come out tails? (b) after how many straight tails can we say that with 90% probability the coin i hold is biased? (c) after n straight tails, what is the probability that the next flip is also tails? (d) suppose we have flipped a very large number of times (think number of flips n tending to infinity), and each time gotten tails. what are the chances that the next flip again yields tails?
Answers
The probability that the coin is a biased coin after one flip that results in tails is about 10.9%.
Bayes' theorem is a mathematical formula that helps us update our beliefs about the probability of an event occurring based on new information or evidence. It involves the prior probability, which is our initial belief about the probability of an event, and the likelihood of the evidence given that the event has occurred. By combining these two pieces of information, Bayes' theorem allows us to calculate the updated or posterior probability of the event occurring.
After one flip that results in tails, the probability that the coin is a biased coin can be found using Bayes' theorem:
P(biased coin | tails) = P(tails | biased coin) * P(biased coin) / P(tails)
P(tails | biased coin) = 3/5 (given)
P(biased coin) = 0.1 (given)
P(tails) = P(tails | fair coin) * P(fair coin) + P(tails | biased coin) * P(biased coin)
= 1/2 * 0.9 + 3/5 * 0.1
= 0.55
Therefore,
P(biased coin | tails) = (3/5 * 0.1) / 0.55
= 0.109 (approximately)
Thus, the probability that the coin is a biased coin after one flip that results in tails is about 10.9%.
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Assume that 90% of the coins in circulation are fair, and the remaining 10% are biased coins that give tails with probability 3/5. i hold a randomly chosen coin and begin to flip it.
(a) after one flip that results in tails, what is the probability that the coin i hold is a biased coin? after two flips that both give tails? after n flips that all come out tails?