the partial fraction decomposition of 40x2−4 can be written in the form of f(x)x−2 g(x)x 2,
Answers
The partial fraction decomposition of 40x^2 - 4 can be written in the form of (10/x) - (10/(x^2/10 + 2)).
To find the partial fraction decomposition, we need to factorize the given expression. First, we can take out a common factor of 4 from both terms, which gives us:
40x^2 - 4 = 4(10x^2 - 1)
We can then factorize the expression inside the brackets using the identity a^2 - b^2 = (a + b)(a - b), where a = sqrt(10)x and b = 1/sqrt(10):
10x^2 - 1 = (sqrt(10)x)^2 - 1^2 = (sqrt(10)x + 1)(sqrt(10)x - 1)
Substituting this back into our original expression, we get:
40x^2 - 4 = 4(10x^2 - 1) = 4(sqrt(10)x + 1)(sqrt(10)x - 1)
Now, we can use the method of partial fractions to write this expression in the given form. We assume that the decomposition has the form:
40x^2 - 4 = A/x + B/(x^2/10 + 2)
Multiplying both sides by the denominator and simplifying, we get:
4(sqrt(10)x + 1)(sqrt(10)x - 1) = Ax^2/10 + 2A + Bx
We can solve for A and B by equating coefficients of like terms on both sides of the equation. First, we let x = 0, which gives us:
4 = 2A
A = 2
Next, we let x^2/10 + 2 = 0, which gives us:
sqrt(10)x = +/- sqrt(20)i
where i is the imaginary unit. Substituting x = sqrt(20)i/sqrt(10) into the equation and simplifying, we get:
-8 = -2A
A = 4
Finally, equating coefficients of x on both sides, we get:
4sqrt(10)B = 0
B = 0
Therefore, the partial fraction decomposition is:
40x^2 - 4 = (10/x) - (10/(x^2/10 + 2))
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Related Questions
A coin is loaded so that the probability of heads is 0.9 and the probability of tails is 0.1. Suppose that the coin is tossed twice and that the results of the tosses are independent. For i = 1 and 2, let H; be the event that the result of toss i is heads, and let T; be the event that the result of toss i is tails. (a) What is the probability (as a %) of obtaining exactly two heads? 81 % (b) What is the probability (as a %) of obtaining exactly one head? The probability of obtaining exactly one head is P((H, NT, U (H, NT,)) v U which equals %. Enter an exact number. (C) What is the probability (as a % of obtaining no heads? % (d) What is the probability of obtaining at least one head? % Use the binomial theorem to expand the expression.
Answers
a) The probability of obtaining exactly two heads is 81%.
(b) The probability of obtaining exactly one head is 18%.
(c) The probability of obtaining no heads is 1%.
(d) The probability of obtaining at least one head is 99%.
(a) The probability of obtaining exactly two heads can be calculated by multiplying the probabilities of getting heads on both tosses. Given that the probability of heads is 0.9, the probability of obtaining two heads is 0.9 * 0.9 = 0.81, or 81%.
(b) To calculate the probability of obtaining exactly one head, we need to consider two cases: (1) heads on the first toss and tails on the second toss, or (2) tails on the first toss and heads on the second toss. The probability of heads on the first toss is 0.9, and the probability of tails on the second toss is 0.1. Multiplying these probabilities gives 0.9 * 0.1 = 0.09. Similarly, the probability of tails on the first toss and heads on the second toss is 0.1 * 0.9 = 0.09. Adding these two probabilities together gives a total probability of 0.09 + 0.09 = 0.18, or 18%.
(c) The probability of obtaining no heads can be calculated by multiplying the probabilities of getting tails on both tosses. Given that the probability of tails is 0.1, the probability of obtaining no heads is 0.1 * 0.1 = 0.01, or 1%.
(d) The probability of obtaining at least one head can be calculated by subtracting the probability of obtaining no heads from 1. In this case, the probability of obtaining no heads is 0.01 (as calculated in part c). Therefore, the probability of obtaining at least one head is 1 - 0.01 = 0.99, or 99%.
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plz help i dont understand this at all answer fast
Answers
Answer:
110!
Step-by-step explanation:
The two shapes are similar which means all of the angles measures are congruent therefore since ADC is 110, then z must also be 110
David ha fabricado una caja de cartón sin tapa para guardar sus cromos la caja tiene una base cuadrada cuyos lados miden 30 cm y cuya altura mide 10 cm dibujando su desarrollo plano calcula la cantidad de cartón que necesito David
Answers
Answer:
wut language is this
Step-by-step explanation:
Suppose that you knew the following compound statement Q⟹(R∧Q) Is false. What can you say about R? R must be true R must be false There is not enough information to determine the truth value of R
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Given a compound statement Q ⟹ (R ∧ Q) is false. The answer to what can we say about R is: R must be false.What are compound statements?Compound statements are also known as a logical statement or a statement. It is defined as a statement formed by joining two or more simple statements using logical operators.A compound statement is made up of simple statements combined using logical operators such as "or", "and", "if-then", and "if and only if."Example: The statement "It is raining and the sun is shining" is a compound statement that contains the simple statements "It is raining" and "The sun is shining," joined by the logical operator "and."What is the given statement?The given statement is: Q ⟹ (R ∧ Q) is false.If we look closely at the statement, we can see that it is a conditional statement because it has the word "if" in it. And we know that the conditional statement is only false when the hypothesis is true, and the conclusion is false.What can we say about R?Since the conditional statement Q ⟹ (R ∧ Q) is false, that means the hypothesis Q is true and the conclusion R ∧ Q is false.If Q is true and R ∧ Q is false, then R must be false because if R is true, then R ∧ Q would be true.Hence, the answer to what can we say about R is: R must be false.
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The graph of f(x) and g(x) are shown below. How many solutions does the system of equations have?
Click pic to see whole problem
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Answer:
Step-by-step explanation:
Solving systems of equations gives the points of intersection when the equations are graphed.
The answer is 3.
Given matrices A of mxn B of pxq C = AB Which of the following statements about C is true? C exists only if n = p C does not exist C exists only if m = q C is always a square matrix
Answers
The statement that is true about matrix C, which is the product of matrices A and B (C = AB), is that C exists only if n = p.
For matrix multiplication to be defined, the number of columns in matrix A must be equal to the number of rows in matrix B. In other words, if A is an mxn matrix and B is a pxq matrix, then the product AB will only exist if the number of columns in A (n) is equal to the number of rows in B (p). If n is not equal to p, then the product AB cannot be calculated, and C does not exist. Therefore, the correct statement is that C exists only if n = p. It is important to note that this condition is necessary for the matrix product to be defined, but the resulting matrix C may or may not be square depending on the dimensions of A and B.
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what are the advantages and disadvantages of using a permutation as in problem 1 over an arbitrary permutation?
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Advantages of using a permutation as in problem 1 include increased efficiency and simplicity, while disadvantages include reduced flexibility and a potential loss of information.
Permutations are mathematical operations that rearrange elements in a list or sequence, and there are different ways to define and generate permutations. A permutation specified in problem 1 may have certain desirable properties, such as being easy to compute, having low computational complexity, or having a simple mathematical form.
These advantages, however, can also be disadvantages if they limit the ability to capture more complex or subtle relationships between elements, or if they prevent the use of other methods that may be more appropriate for a particular application. Therefore, the choice of a specific permutation method depends on the requirements of the problem and the available computational resources.
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The following points for a function are graphed in the coordinate plane: (1, 2), (3, 8), (4, 11), and (6, 17).
What function is being graphed?
5x - 3
y = 3x - 1
y = x + 1
y = 2x
Answers
I upload I picture of showing my work. Hope this help you.
Answer: y=3x-1 is your answer
Step-by-step explanation:
now I have to add stuff
A family has two children. What is the probability that both the children are boys given that at least one of them is a boy?
Answers
Answer:
the probability is 1/2
Step-by-step explanation:
the probability of a child being a boy is 1/2, if at least one of them is already a boy, then the probability of both being boys, which amounts to saying that the other is also a boy, is 1/2
ill give 30
points for this whole thing help quick
Answers
Answer:
33
Step-by-step explanation:
i don't know 1 2 and 3 but here are the rest
4) 7 - 4
5) -140 + 112
6) 18 - 4 + 12
7) 72 - 12
8) 1
9) 5
10) -14
11) 8
12) -26
13) -5
14) 61
15) -25
16) 11
17) -47
18) -14
19) 11
20) -7
21) 5
22) 8
23) -1
24) 22
25) 1
26) -16
27) 44
28) -1
29a) Jill
29b) Jill needs 4 points to win and Joe needs 6 points.
what is the number of the parking space 16, 06, 68
Answers
The number formed by the digits 16, 06, and 68 is 160668, which is determined by concatenating them in the given order.
To determine the number formed by the given digits, we concatenate them in the given order. Starting with the first digit, we have 16. The next digit is 06, and finally, we have 68. By combining these three digits in order, we get the number 160668.
When concatenating the digits, the position of each digit is crucial. The placement of the digits determines the resulting number. In this case, the digits are arranged as 16, 06, and 68, and when they are concatenated, we obtain the number 160668. It's important to note that the leading zero in the digit 06 does not affect the value of the resulting number. When combining the digits, the leading zero is preserved as part of the number.
Therefore, the number formed by the digits 16, 06, and 68 is 160668.
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Is \(\frac{x^{2}}{2}\) an equation that can be simplified? If so, what is its simplified form?
Answers
hope this helped you
Can someone make sure that the first one is right and can someone help me with the second one please I need to get this right please
Answers
Answer:
1. B
2. A
Step-by-step explanation:
1. Here, the figure is reflected over the line depicted. This answer is correct.
2. Here the image shifts down and across, making it a glide.
Many thermal energy conversion devices, such as thermoelectric, AMTEC and Stirling converters developed for space applications, also have terrestrial uses. Rather than use radioisotopes as a heat, a combustion flame can be used. Consider a converter producing 100 We with a 15% thermal to electric conversion efficiency designed to produce power continuously for one year. (a) What mass of 238Pu would be needed? (b) What mass of a petroleum-based fuel, with density 0.9 g/cm3 and a heat of combustion of 40 MJ/L, would be needed?
Answers
To determine the mass of 238Pu and the mass of a petroleum-based fuel needed for the thermal energy conversion device, we need to consider the power output and efficiency of the converter.
(a) The converter produces 100 We with a 15% thermal to electric conversion efficiency. To calculate the mass of 238Pu needed, we need to know the energy density of 238Pu. Once we have that information, we can divide the power output by the efficiency to obtain the thermal energy input. Then, we can divide the thermal energy input by the energy density of 238Pu to get the mass.
(b) To calculate the mass of the petroleum-based fuel needed, we can use the heat of combustion and the power output of the converter. First, we need to convert the heat of combustion from MJ/L to J/cm3 by multiplying it by 10^6 and dividing by the density. Then, we divide the power output by the heat of combustion per unit volume to obtain the volume of fuel needed. Finally, we multiply the volume by the density to get the mass.
Remember to use the appropriate unit conversions and formulas to accurately calculate the masses of 238Pu and the petroleum-based fuel.
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the heights, in inches, of orangutans are normally distributed with an unknown population mean. a random sample results in a 95% confidence interval of (52,58) inches. what is the correct interpretation of the 95% confidence interval? select the correct answer below: we estimate with 95% confidence that the sample mean is between 52 and 58 inches. we estimate that 95% of orangutans have a height between 52 and 58 inches. we estimate with 95% confidence that the true population mean is between 52 and 58 inches.
Answers
The correct interpretation of the 95% confidence interval is:
"We estimate with 95% confidence that the true population mean height of orangutans is between 52 and 58 inches."
What is statistics?
Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data. It involves the use of methods and techniques to gather, summarize, and draw conclusions from data.
The confidence interval provides a range of values within which the true population mean height is likely to fall with a 95% level of confidence. It does not provide information about individual orangutans' heights or the sample mean's precise location within the interval.
Therefore, The correct interpretation of the 95% confidence interval is:
"We estimate with 95% confidence that the true population mean height of orangutans is between 52 and 58 inches."
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PLEASE HELP I have no idea what I'm doing!!! I'm giving 20 points!!
Answers
Answer:
A
B
C
Step-by-step explanation:
100% correct answers
Answer:
C
Step-by-step explanation:
Which one of the following procedures would improve the reliability and validity of grading shortessay tests, thus refuting the complaint of sensitivity to bias and variability in grading?A) Administering more pretestsB) Grading on the curveC) Implementing a contract systemD) Using a scoring rubric
Answers
The most appropriate procedure from the given options that would improve the reliability and validity of grading short essay tests, thus addressing the complaint of sensitivity to bias and variability in grading, is: D) Using a scoring rubric.
A scoring rubric is a predefined set of criteria or guidelines that outlines the specific expectations and standards for evaluating student responses. By using a scoring rubric, the grading process becomes more objective, consistent, and transparent, thereby reducing bias and variability in grading.
A scoring rubric provides a structured framework that allows graders to assess and assign scores based on predetermined criteria, such as content knowledge, organization, clarity, coherence, and grammar. It ensures that all essays are evaluated using the same standards and eliminates subjective judgment as much as possible.
Administering more pretests (option A) may provide additional practice or feedback to students but does not directly address the issue of bias and variability in grading.
Grading on the curve (option B) adjusts grades based on the performance of other students in the class, which does not directly address bias or variability in grading. It can introduce its own set of issues and may not reflect the true quality of each individual student's work.
Implementing a contract system (option C) refers to a negotiated agreement between students and instructors regarding grading criteria, which may help clarify expectations but does not inherently address bias or variability.
Therefore, using a scoring rubric (option D) is the most appropriate procedure to improve the reliability and validity of grading short essay tests while minimizing sensitivity to bias and variability in grading.
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Find the equation of the line passing through the given point and parallel to the given line.
Answers
Answer:
\(y-2=-\frac{1}{2}(x-3)\)
Step-by-step explanation:
\(x+2y=5 \\ \\ 2y=-x+5 \\ \\ y=-\frac{1}{2}x+\frac{5}{2}\)
The slope of the given line is \(-\frac{1}{2}\).
This means the slope of any line parallel to the given line is also \(-\frac{1}{2}\).
Using point-slope form, the equation is \(y-2=-\frac{1}{2}(x-3)\).
is the integer r divisible by 3 ? (1) r is the product of 4 consecutive positive integers. (2) r < 25
Answers
Yes, the integer is divisible by 3.
To determine if r is divisible by 3, we must first determine the value of r. According to statement 1, r is the product of 4 consecutive positive integers.
This means that the 4 integers must be increasing by 1 each time. The smallest integer must be 1, and the 4th integer must be 4 higher than the first, making it 5.
The product of these 4 integers is 1 x 2 x 3 x 4 = 24, which is less than 25.
Since 24 is divisible by 3, r is also divisible by 3.
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Duncan is putting icing on the top of a rectangular cake if the top of the cake is 6 inches wide by 10 inches long what is the area of the top of the cake
A. 80 square inches
B.60 square inches
C.30 square inches
D. 120 square inches
Answers
Duncan is putting icing on the top of a rectangular cake if the top of the cake is 6 inches wide by 10 inches long what is the area of the top of the cake
A. 80 square inches
B.60 square inches
C.30 square inches
D. 120 square inches
Answer:
B- 60 square inches
Hari's weekly allowance varies depending on the number of chores he does. He received $20 in allowance the week he did 14 chores, and $14 in allowance the week he did 2 chores. Write an equation for his allowance in slope-intercept form. The equation for Hari's allowance in slope-intercept form is
Answers
explain better or brainliest cant do that
The rational number -13/-5 lies to the right of zero on the number true or false.
Answers
Answer:
true
hope it helps
PLEASE MARK BRAINLIEST
(Equivalent Algebraic Expressions LC)
Simplify this please!!!
Answers
Answer: X^3 / Y^5
The first answer choice
Step-by-step explanation:
614 divided by 15??????????
Answers
Answer:
122.8 brainliest plzzz
Step-by-step explanation:
Answer:
040
1 5 6 1 4
− 0
6 1
− 6 0
1 4
− 0
1 4
Step-by-step explanation:
40 reminder 14 is the answer
Write two fractions equivalent to 4
_
12
Answers
Answer:
2/6
1/3
Step-by-step explanation:
the residents of a city voted on whether to raise property taxes. the ratio of yes votes to no votes was to . if there were no votes, what was the total number of votes?
Answers
The total number of no votes are 4422.
What is ratio?The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity.
Given: The ratio of yes votes to no votes is 5 to 6 i.e.
yes votes /no votes = 5/6
It means there are total 11 votes ( 5 votes for yes and 6 votes for no)
which gives
no votes/total votes= 6/11---(i)
But in the given question total votes are 8107,
so let us consider there are x votes for no which implies
no votes/total votes= x/8107
From (i), we have
x/8107 = 6/11
⇒ x = 6*8107/11 = 4422.
Then the total number of no votes is 4422.
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NOTE: The question given in the portal is incomplete. Here is the complete question.
Question: The residents of a city voted on whether to raise property taxes. The ratio of yes votes to no votes was 5 to 6. If there were 8107 total votes, how many no votes were there?
A rock sample containing an isotope with a half-life of 28 million years has an initial mass of 184 grams. how much time has elapsed after three half-lives?
Answers
The half life period is the time in which only half of the given population remains. It can be represented through this equation:
\(f(t)=a\times(1/2)^{\frac{t}{h}}\)
t = time passeda = y-intercepth = half lifeSolving the QuestionWe're given:
h = 28 million yearsa = 184 grams (this is the initial mass, after 0 time has passed)For most questions like this, we would have to plug these values into the equation mentioned above. However, this question asks for the time elapsed after 3 half-lives.
This can be calculated simply by multiplying the given half-life by 3:
28 million years x 3
= 84 million years
Answer84 million years
find the value of w. round to the nearest tenth
Answers
Answer:
\(\pmb {w=13.11}\)Step-by-step explanation:
\(\pmb {sin(22)^o=\cfrac{x}{35} }\)
\(\pmb {35sin(22)=w}\)
\(\pmb {w=13.11}\)
_________________
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Julian is making potato salad for his family picnic. At the market, he spends $5.58 on red potatoes and $4.68 on yellow potatoes. If each type of potato cost $0.90 per pound, how many total pounds of potatoes does he buy
Answers
Answer:
11.4 pounds
Step-by-step explanation:
Prove by Mathematical Induction that 11^n + 2 + 12^2n + 1 is divisible by 133 Vn ≥1
Answers
We will prove by mathematical induction that for all n ≥ 1, the expression 11^n + 2 + 12^(2n + 1) is divisible by 133. The base case is n = 1, where the expression evaluates to 137, which is divisible by 133. We assume that the expression is divisible by 133 for some value k = m, and then prove that it holds true for k = m + 1.
Base case: For n = 1, the expression becomes 11^1 + 2 + 12^(2*1 + 1) = 11 + 2 + 12^3 = 137. Since 137 is divisible by 133 (137 = 1 * 133 + 4), the base case is true.Inductive step: Assume that for some value k = m, the expression 11^m + 2 + 12^(2m + 1) is divisible by 133, i.e., it can be expressed as 133a, where a is an integer. Now we need to prove that for k = m + 1, the expression 11^(m + 1) + 2 + 12^(2(m + 1) + 1) is divisible by 133.
Expanding the expression, we have 11^(m + 1) + 2 + 12^(2m + 3) = 11 * 11^m + 2 + 12 * 12^(2m + 1).Using the assumption, we can write this expression as (133a - 2) * 11 + 2 + 12 * 12^(2m + 1). Simplifying further, we have 133a * 11 + 12 * 12^(2m + 1).Factoring out 133, we get 133(a * 11 + 12^(2m + 1)).
Since both a * 11 and 12^(2m + 1) are integers, the expression is divisible by 133.Therefore, by mathematical induction, we have proven that for all n ≥ 1, the expression 11^n + 2 + 12^(2n + 1) is divisible by 133.
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