which ordered pairs are are solutions to the equation 5x 6y=13? select all that apply: (−1,3) (3,−1/3) (3,−2) (7,−1) none of the above

By the principle of mathematical induction, we can conclude that for all  n ≥ 0, bn = 2n+1 - 1.

To prove that for n ≥ 0, bn = 2n+1 - 1, we will use mathematical induction.

Base case: When n = 0, we have b0 = 1, and 2(0) + 1 - 1 = 0, which satisfies the given formula.

Induction hypothesis: Assume that for some integer k ≥ 0, we have bk = 2k+1 - 1.

Induction step: We will prove that if the induction hypothesis is true for k, then it is also true for k + 1. That is, we will show that bk+1 = 2(k+1)+1 - 1.

Using the recurrence relation given in the problem statement, we have:

bk+1 = 2bk + 1

= 2(2k+1 - 1) + 1 (by the induction hypothesis)

= 2(2k+1) - 1

= 2(k+1)+1 - 1

Therefore, we have shown that if the induction hypothesis is true for k, then it is also true for k + 1. By the principle of mathematical induction, we can conclude that for all n ≥ 0, bn = 2n+1 - 1.

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the probability that there is at most one purple marble when three marbles are randomly chosen from the bag is approximately 0.770.

To find the probability that there is at most one purple marble when three marbles are randomly chosen from the bag, we need to consider the different scenarios:

Scenario 1: No purple marbles are chosen

In this case, we can choose 3 marbles from the remaining yellow, green, and red marbles. The number of ways to choose 3 marbles from a set of 22 marbles (8 yellow + 9 green + 5 red) is given by the combination formula: C(22, 3).

Scenario 2: One purple marble is chosen

In this case, we need to choose 2 marbles from the remaining yellow, green, and red marbles, and 1 marble from the 3 purple marbles. The number of ways to choose 2 marbles from 22 marbles and 1 marble from 3 marbles is given by the combination formula: C(22, 2) * C(3, 1).

The total number of ways to choose 3 marbles from the 25 marbles in the bag (8 yellow + 9 green + 3 purple + 5 red) is given by: C(25, 3).

To find the probability, we sum the probabilities of both scenarios and divide by the total number of ways to choose 3 marbles:

Probability = (Number of ways for scenario 1 + Number of ways for scenario 2) / Total number of ways

Probability = (C(22, 3) + (C(22, 2) * C(3, 1))) / C(25, 3)

Using a calculator or computer program to calculate the combinations, we can find:

Probability ≈ 0.770

Therefore, the probability that there is at most one purple marble when three marbles are randomly chosen from the bag is approximately 0.770.

The correct answer is 0.770, corresponding to option 0.770.

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≈$5,905.67

Total Interest: $1,905.67

[tex]A=P(1+\frac{r}{n} )^{nt}[/tex] where:

[tex]A[/tex] = final amount,

[tex]P[/tex] = initial principal: 4000 ,

[tex]r[/tex] = interest rate: 4.9%,

[tex]n[/tex] = number of times interest applied per time period: quarterly; 4

and [tex]t[/tex] = time: in years; 8

thus:

[tex]A=4000(1+\frac{0.049}{4} )^{32}[/tex]

Suppose 60% of the area under the standard normal curve lies to the right of z. The value of z is greater than zero. This statement is True.

We know that the standard normal distribution is symmetric.

So, if we divide the area of the curve into two parts, each part will have 50% area. The standard normal distribution is shown below : Now, it is given that 60% of the area under the standard normal curve lies to the right of z. This implies that the remaining 40% area lies to the left of z. Therefore, z is negative because it lies to the left of the mean.

However, it is given that the value of z is greater than zero. This is not possible.

Hence, the given statement is false. However, if the statement was changed to say that 60% of the area lies to the left of z, then the statement would be true. This is because z is a positive value and it lies to the left of the mean.

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Answer:

y = x - 5

Step-by-step explanation:

x is input

y is output

output, y, is 5 less than input, x

y = x - 5

Using Laplace transform, The solution to the initial value problem          y'' + 2y' + 2y = g(t), y(0) = 0, y'(0) = 1, expressed as a piecewise defined function, is:

For π ≤ t < 2π:

y(t) = e^(-t) sin(t)

For t ≥ 2π:

y(t) = 0

To solve the initial value problem using Laplace transforms, we'll apply the Laplace transform to both sides of the differential equation.

Taking the Laplace transform of the equation  [tex]y'' + 2y' + 2y = g(t)[/tex], we get:

[tex]s^2Y(s) - sy(0) - y'(0) + 2(sY(s) - y(0)) + 2Y(s) = G(s)[/tex]

Applying the initial conditions y(0) = 0 and y'(0) = 1, we have:

[tex]s^2Y(s) - s(0) - 1 + 2(sY(s) - 0) + 2Y(s) = G(s)\\\\s^2Y(s) + 2sY(s) + 2Y(s) - 1 = G(s)[/tex]

Simplifying further, we get:

[tex]Y(s) = G(s) / (s^2 + 2s + 2)[/tex]

Next, we'll find the inverse Laplace transform of Y(s) using partial fraction decomposition. We need to express the denominator as a product of linear factors:

[tex]s^2 + 2s + 2 = (s + 1)^2 + 1[/tex]

The roots of the denominator are -1 ± i. Therefore, we can rewrite Y(s) as:

[tex]Y(s) = G(s) / ((s + 1)^2 + 1)[/tex]

Now, we can take the inverse Laplace transform of Y(s):

[tex]y(t) = L^(-1)[Y(s)] = L^(-1)[G(s) / ((s + 1)^2 + 1)]\\[/tex]

Since g(t) is piecewise defined, we need to split the inverse Laplace transform into two parts based on the intervals of g(t):

For π ≤ t < 2π:

[tex]y(t) = L^(-1)[1 / ((s + 1)^2 + 1)][/tex]

For t ≥ 2π:

y(t) = 0

Now, we need to find the inverse Laplace transform of 1 / ((s + 1)² + 1). Using Laplace transform table properties, we have:

[tex]L^(-1)[1 / ((s + 1)^2 + 1)] = e^(-t) sin(t)[/tex]

Therefore, the solution to the initial value problem y'' + 2y' + 2y = g(t), y(0) = 0, y'(0) = 1, expressed as a piecewise defined function, is:

For π ≤ t < 2π:

y(t) = e^(-t) sin(t)

For t ≥ 2π:

y(t) = 0

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96% percentage is the difference between 17/100 x 20 and 17/20 x 100

The difference between (17/100) x 20 and (17/20) x 100 can be calculated by finding the absolute difference between the two values and expressing it as a percentage of the larger value.

First, let's calculate each expression:

(17/100) x 20 = 0.17 x 20 = 3.4

(17/20) x 100 = 0.85 x 100 = 85

The difference between these two values is |85 - 3.4| = 81.6.

To express this difference as a percentage of the larger value, we divide 81.6 by the larger value (85 in this case) and multiply by 100:

(81.6 / 85) x 100 = 96%

Therefore, the difference between (17/100) x 20 and (17/20) x 100 is approximately 96% of the larger value.

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The statement that can be used to find the value of xy is D. cos m< x = xy/y. Explanation: Let us see what we are given and what we need to find.

Given: A xyz is a triangle with x = 20.5, y = 11.8, and[tex]m < x = 55.4[/tex]°We need to find: Value of xy Step-by-step explanation: In a right triangle, the cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse. [tex]cos m < x = xy/y cos 55.4 = xy/20.5xy = 20.5 × cos 55.4 = 20.5 × 0.5736 ≈[/tex]11.76Therefore, the value of xy is approximately 11.76.

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To divide the polynomial (5x^2 + 14x + 2)^2 - (4x^2 - 5x + 7)^2 by the polynomial x^2 + x + 1, we can use polynomial long division. The divisor x^2 + x + 1 is a quadratic polynomial, so we divide the polynomial into the leading terms of the dividend.

Performing the long division, we divide (5x^2 + 14x + 2)^2 - (4x^2 - 5x + 7)^2 by x^2 + x + 1. The quotient obtained will be the quotient q, and the remainder obtained will be the remainder r.

After completing the long division, we can express the quotient and remainder in terms of the divisor x^2 + x + 1. The quotient q will be a polynomial, and the remainder r will be a polynomial divided by the divisor.

To divide (5x^2 + 14x + 2)^2 - (4x^2 - 5x + 7)^2 by x^2 + x + 1, we use polynomial long division. The quotient q is the result of the division, and the remainder r is the remainder obtained after the division. Both q and r are expressed in terms of the divisor x^2 + x + 1.

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Expression C (4,480 * 1.008x) cannot be used to determine Marissa's net proceeds because it does not consider the broker fee or the online broker fee, which should be deducted from the final proceeds.

Let's evaluate each expression to determine which one cannot be used to determine Marissa's net proceeds.

A. 4,500 * 1.008x - 20

This expression represents the final proceeds after deducting the broker fee of 0.8% (0.008) and the online broker fee of $20. It correctly calculates the net proceeds and can be used.

B. 4,500 - (0.08x + 20 + x)

This expression subtracts various fees (broker fee and online broker fee) and the initial investment amount from the final stock price. It correctly calculates the net proceeds and can be used.

C. 4,480 * 1.008x

This expression multiplies the stock price before deducting any fees by the investment amount. However, it does not account for the broker fee or the online broker fee, which should be subtracted from the final proceeds. Therefore, this expression cannot be used to determine Marissa's net proceeds.

D. 4,500 * (1.008x + 20)

This expression multiplies the stock price after deducting the online broker fee by the investment amount and the broker fee. It correctly calculates the net proceeds and can be used.

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Complete Question:

Marissa purchased x dollars worth of stock and paid her broker a 0.8% fee. She sold the stock when the stock price increased to $4,500 using an online broker that charged $20 per trade.

Which expression below cannot be used to determine her net proceeds?

A. 4,500 1.008x-20

B. 4,500-(0.08x+20+x)

C. 4,4801.008x

D. 4,500 (1.008x+20)

The statement "A low value of the correlation coefficient r implies that x and y are unrelated" is false.

In the context of correlation coefficient (r), the value of r measures the strength and direction of the linear relationship between two variables, x and y. It ranges from -1 to +1, where -1 indicates a perfect negative linear relationship, +1 indicates a perfect positive linear relationship, and 0 indicates no linear relationship.

A low value of the correlation coefficient (close to 0) does not necessarily imply that x and y are unrelated. It only suggests that there is a weak linear relationship between the variables. However, it is important to note that there could still be other types of relationships or associations between the variables that are not captured by the correlation coefficient.

Therefore, a low value of the correlation coefficient does not provide definitive evidence that x and y are unrelated. It is necessary to consider other factors, such as the nature of the data, the context of the variables, and potential nonlinear relationships, before concluding whether x and y are truly unrelated.

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The measurement of the angle MRX is 130°.

Given that a figure we need to find the angle MRX,

The lines TP and ZX are perpendicular to each other, and there is a line MQ intersecting at R,

So,

Angles MRT and MRZ are complementary so,

m ∠MRZ + m ∠MRT = 90°

50° + m ∠MRT = 90°

m ∠MRT = 40°

Also,

Angles TRX and TRZ are supplementary so, and equal to right angle, so,

m ∠MRX = m ∠MRT + m ∠TRX

m ∠MRX = 90° + 40°

m ∠MRX = 130°

Hence the measurement of the angle MRX is 130°.

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The equation of the line passing through (-2,1) and (-2,0) is x = -2.

:Given two points (-2,1) and (-2,0), to find the equation of the line passing through these points. Use the following steps;Find the slope of the line using the formula;y2 - y1 / x2 - x1

Simplify the equation of the slope and plug in any point.Find the equation in slope-intercept form by using the point-slope formThe formula of the slope is;Δy / Δx = (y2 - y1) / (x2 - x1)Let the points (-2,1) and (-2,0) be (x1,y1) and (x2,y2) respectively.

Summary:Therefore, option A is the correct answer which is x = -2, as the equation of the line passing through (-2,1) and (-2,0).

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The fraction that represents a repeating decimal when converted is given as follows:

2/11.

A fraction is represented by the division of a term x by a term y, such as in the equation presented as follows:

Fraction = x/y.

The terms that represent x and y are listed as follows:

The decimal representation of each fraction is given by the division of the numerator by the denominator, hence:

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The probability that the sample mean is greater than 168 is approximately 0.0655, or 6.6% (rounded to the nearest tenth of a percent).

To find the probability that the sample mean is greater than 168, we can use the central limit theorem and the properties of the normal distribution.

The central limit theorem states that for a large enough sample size (in this case, n = 50), the distribution of sample means will approach a normal distribution, regardless of the shape of the population distribution.

Given that the population mean is 164 and the population standard deviation is 18.65, we can calculate the standard deviation of the sample mean, also known as the standard error, using the formula:

Standard Error (SE) = Population Standard Deviation / √(Sample Size)

SE = 18.65 / √50

SE ≈ 2.636

Next, we need to standardize the value of 168 using the sample mean and the standard error. This allows us to calculate the probability using the standard normal distribution.

Z = (Sample Mean - Population Mean) / Standard Error

Z = (168 - 164) / 2.636

Z ≈ 1.516

To find the probability that the sample mean is greater than 168, we can look up the corresponding area under the standard normal curve to the right of Z = 1.516. This can be done using a standard normal distribution table or a statistical calculator.

Using a standard normal distribution table, we find that the area to the right of Z = 1.516 is approximately 0.0655.

Therefore, the probability that the sample mean is greater than 168 is approximately 0.0655, or 6.6% (rounded to the nearest tenth of a percent).

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To determine a basis for the set spanned by the given vectors, we can perform row operations on the augmented matrix [v1 | v2 | v3 | v4 | v5 | v6] and identify the pivot columns.

Row-reducing the augmented matrix yields:

[1 3 1 5 2 2 | 0]

[2 6 3 11 7 0 | 0]

[3 9 5 17 12 0 | 0]

By performing row operations, we can simplify the matrix to its row-echelon form:

[1 3 1 5 2 2 | 0]

[0 0 1 1 3 0 | 0]

[0 0 0 0 0 0 | 0]

The pivot columns are the columns with leading 1's in the row-echelon form. In this case, the pivot columns are 1, 3, and 5.

Therefore, a basis for the set spanned by the given vectors is {v1, v3, v5}, which corresponds to the columns of the original matrix in the pivot columns. These three vectors are linearly independent and can span the entire space represented by the given vectors.

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There are 10 different choices of pens with this brand

To find out how many different choices of pens you have with a popular brand of pen available in 5 colors and 2 writing tips, you can use the multiplication principle of counting.

The multiplication principle of counting states that if there are m ways to do one thing, and n ways to do another, then there are m * n ways of doing both.

This principle applies even if there are more than two things to consider.

Hence, to solve this problem, you can simply multiply the number of colors by the number of writing tips as follows:

5 colors × 2 writing tips = 10

Therefore, there are 10 different choices of pens with this brand.

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The most appropriate model to represent the data in the table is quadratic

From the question, we have the following parameters that can be used in our computation:

The graph

In the graph, we can see that

Only a quadratic function has this feature

Hence, the most appropriate model to represent the data in the table is quadratic

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There are 792 poker hands consisting of all face cards.

To determine the number of poker hands consisting of all face cards, we need to consider the number of ways we can select 5 face cards from the 12 available face cards.

Since we are selecting a specific number of items from a larger set without considering the order, we can use combinations to calculate the number of poker hands.

The number of combinations of selecting k items from a set of n items is given by the formula:

C(n, k) = n! / (k!(n-k)!)

In this case, we want to select 5 face cards from the set of 12 face cards, so we can calculate:

C(12, 5) = 12! / (5!(12-5)!)

C(12, 5) = 12! / (5! * 7!)

Calculating the factorial terms:

12! = 12 * 11 * 10 * 9 * 8 * 7!

5! = 5 * 4 * 3 * 2 * 1

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

Plugging in the values:

C(12, 5) = (12 * 11 * 10 * 9 * 8 * 7!) / (5 * 4 * 3 * 2 * 1 * 7!)

Simplifying the expression:

C(12, 5) = (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1)

C(12, 5) = 792

Therefore, there are 792 poker hands consisting of all face cards.

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The inverse function of C(U) = U + 15 is U(C) = C - 15, representing the uncurved grade in terms of the curved grade. The correct option is OD. U(C) = C - 15.

To find the inverse function of C(U) = U + 15, we need to switch the roles of U and C and solve for U.

Let's denote the inverse function as U(C).

C = U + 15

To find U, we subtract 15 from both sides:

C - 15 = U

Therefore, the inverse function is U(C) = C - 15.

Among the given options, the correct inverse function is OD. U(C) = C - 15.

This inverse function represents the uncurved grade (U) in terms of the curved grade (C). It allows us to determine the original uncurved grade when we know the curved grade after applying a curve of adding 15.

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If the probability of success is 0.730, the value of log odds is 0.994 when rounded to the thousandths place. What is the Log Odds ratio?

The odds ratio is defined as the ratio of the probability of success to the probability of failure:[tex]$$OR = \frac{p}{1-p}$$T$$\ln(OR) = \ln \frac{p}{1-p}$$.$$\ln \frac{p}{1-p} = \ln \frac{0.73}{1-0.73}$$$$\ln \frac{p}{1-p} = \ln \frac{0.73}{0.27}$$$$\ln \frac{p}{1-p} = 0.994$$[/tex]

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The missing numbers can be filled up as follows:

1. 200

2. 20%

3. 225

4. 800

5. 2%

To fill up the table, note that percentage is obtained by dividing a base by rate. The rate will also be changed to the decimal format before the computation is done. On this note:

P = B * R

1. 20 = x * 0.1

20 = 0.1x

Divide both sides by 0.1

x = 200

2. 90 = 450 * R

R = 90/450

R = 0.2 OR 20%

3. P = 900 * 0.25

P = 225

4. 280 = B * 0.35

B = 280/0.35

B = 800

5. 14 = 700 * R

R = 14/700

R = 0.02 OR 2%

So, with the given formula, we could generate the base, rate, and percentages of the numbers.

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Answer:

To solve for x in the equation 2(x + 1) = -8, we can use the following steps:

Distribute the 2 on the left side of the equation:

2x + 2 = -8

Subtract 2 from both sides to isolate the x term:

2x = -10

Divide both sides by 2 to solve for x:

x = -5

Therefore, the solution for x is -5.

Answer:

x=-5

Step-by-step explanation:

multiple 2 by x and 1

2x+2

then subtract 2 on both sides

2x=-10

divide 2x from both sides

x=-5

(a)  the probability of obtaining a mean cholesterol level no more than 208 mg/dL is 0.50 or 50%.

(b) the probability of the mean cholesterol level being between 203 and 213 mg/dL.

(c) The probability will give us the likelihood of obtaining a mean cholesterol level less than 198 mg/dL.

(d) the probability that the mean cholesterol level of the sample will be greater than 217 is 15.1%.

a. The probability that the mean cholesterol level of the sample will be no more than 208 mg/dL can be calculated using the z-score formula. First, we need to calculate the z-score for 208 mg/dL, which is (208 - 208) / (35 / √47) = 0. The z-score of 0 corresponds to the mean, and since the cholesterol levels are normally distributed, the probability of obtaining a mean cholesterol level no more than 208 mg/dL is 0.50 or 50%.

b. To calculate the probability that the mean cholesterol level of the sample will be between 203 and 213 mg/dL, we need to calculate the z-scores for both values. The z-score for 203 mg/dL is (203 - 208) / (35 / √47) ≈ -0.7143, and the z-score for 213 mg/dL is (213 - 208) / (35 / √47) ≈ 0.7143. Using a standard normal distribution table or calculator, we can find the probability associated with each z-score. Subtracting the probability associated with the lower z-score from the probability associated with the higher z-score gives us the probability of the mean cholesterol level being between 203 and 213 mg/dL.

c. To calculate the probability that the mean cholesterol level of the sample will be less than 198 mg/dL, we need to calculate the z-score for 198 mg/dL. The z-score is (198 - 208) / (35 / √47) ≈ -1.7143. Again, using a standard normal distribution table or calculator, we can find the probability associated with this z-score. The probability will give us the likelihood of obtaining a mean cholesterol level less than 198 mg/dL.

d. To find the probability that the mean cholesterol level of the sample will be greater than 217 mg/dL, we calculate the z-score for 217 mg/dL: (217 - 208) / (35 / √47) ≈ 1.03. Using the standard normal distribution table or calculator, we find the area to the right of this z-score, which corresponds to the probability. The probability is approximately 0.151 or 15.1%.

Complete Question:

According to a recent poll, 28% of adults in a certain area have high levels of cholesterol. They report that such elevated levels "could be financially devastating to the regions healthcare system" and are a major concern to health insurance providers. Assume the standard deviation from the recent studies is accurate and known. According to recent studies, cholesterol levels in healthy adults from the area average about 208 mg/dL, with a standard deviation of about 35 mg/dL, and are roughly Normally distributed. If the cholesterol levels of a sample of 47 healthy adults from the region is taken, answer parts (a) through (d). a. What is the probability that the mean cholesterol level of the sample will be no more than 208​?

b. What is the probability that the mean cholesterol level of the sample will be between 203 and 213

c. what is the probability that the mean cholesterol level of the sample will be less than 198?

(d) What is the probability that the mean cholesterol level of the sample will be greater than 217?

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The general solution to the second differential equation for y > 0 is (1/3)y^3 = x^2 + 3x + C

To find the general solution to the differential equation (dy)/(dx) = (x-1)/(3y^2) for y > 0, we can separate the variables and integrate.

For the first differential equation:

(dy)/(dx) = (x-1)/(3y^2)

We can rewrite it as:

(3y^2) dy = (x-1) dx

Now we integrate both sides:

∫(3y^2) dy = ∫(x-1) dx

Integrating, we get:

y^3 = (1/2)x^2 - x + C

Where C is the constant of integration.

This is the general solution to the differential equation for y > 0.

For the second differential equation:

(dy)/(dx) = (2x+3)/(y^2)

We can follow the same steps as before:

y^2 dy = (2x+3) dx

Integrating, we get:

(1/3)y^3 = x^2 + 3x + C

Where C is the constant of integration.

This is the general solution to the second differential equation for y > 0.

In both cases, the constant of integration represents the family of all possible solutions to the differential equation.

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Answer: [tex]-1 +\sqrt{7} i[/tex]    and    [tex]-1 -\sqrt{7} i[/tex]

Step-by-step explanation:

[tex]x^2 + 2x +8 = 0[/tex]

We cant factor. so use the quadratic formula and get:

[tex]x = \frac{-2 + \sqrt{-28} }{2}[/tex]        and        [tex]x = \frac{-2 - \sqrt{-28} }{2}[/tex]

these can be simplified to:

[tex]-1 +\sqrt{7} i[/tex]          and            [tex]-1 -\sqrt{7} i[/tex]

and thats it!

For every positive integer n and any real number x, (1 + x)^(2n) ≥ 1 + 2nx.

To prove that for every positive integer n, (1 + x)^(2n) ≥ 1 + 2nx for any real number x, we can use mathematical induction.

Base Case (n = 1):

When n = 1, we need to show that (1 + x)^(2*1) ≥ 1 + 2x.

Simplifying the left side:

(1 + x)^2 = (1 + x)(1 + x) = 1 + 2x + x^2

Comparing it with the right side:

1 + 2x + x^2 ≥ 1 + 2x

Since x^2 ≥ 0 for any real number x, the inequality holds true. So the base case is verified.

Inductive Hypothesis:

Assume that for some positive integer k, the statement holds true, i.e., (1 + x)^(2k) ≥ 1 + 2kx.

Inductive Step:

Now, we need to prove that the statement holds for k + 1, assuming it holds for k.

We start with the left side:

(1 + x)^(2(k+1)) = (1 + x)^(2k + 2) = (1 + x)^2 * (1 + x)^(2k)

Expanding and simplifying the expression:

(1 + x)^2 * (1 + x)^(2k) = (1 + 2x + x^2) * (1 + x)^(2k)

Next, we compare it with the right side:

1 + 2(k+1)x + (k+1)x^2

We can rewrite (k+1)x^2 as kx^2 + x^2.

So now we have:

(1 + 2x + x^2) * (1 + x)^(2k) ≥ 1 + 2(k+1)x + kx^2 + x^2

Expanding further:

(1 + 2x + x^2) * (1 + x)^(2k) ≥ 1 + 2(k+1)x + kx^2 + x^2

By the inductive hypothesis, we know that (1 + x)^(2k) ≥ 1 + 2kx.

Substituting this into the inequality, we have:

(1 + 2x + x^2) * (1 + 2kx) ≥ 1 + 2(k+1)x + kx^2 + x^2

Expanding and simplifying:

1 + 2(k+1)x + 2kx + 4kx^2 + x^2 + 2x^3 + x^2 ≥ 1 + 2(k+1)x + kx^2 + x^2

Now, we can cancel out terms and rearrange to get:

2x^3 + 4kx^2 ≥ kx^2

Since 2x^3 ≥ 0 and 4kx^2 ≥ 0 for any real number x, this inequality holds true.

Therefore, we have shown that if the statement holds for k, it also holds for k+1.

By mathematical induction, we have proven that for every positive integer n, (1 + x)^(2n) ≥ 1 + 2nx for any real number x.

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The intercepts of the graph are x-intercept = (-1, 0) and y-intercept = (0, 2)

From the question, we have the following parameters that can be used in our computation:

The graph

The intercepts of the graph are the points where the graph intersect with the x and the y axes

Using the above as a guide, we have the following:

From the graph, we have the following readings

x-intercept = (-1, 0)

y-intercept = (0, 2)

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The value of x in the equation x degree + x degrees + 90 degrees + x/2 degree = 360 degrees is 108.

In order to solve for x in the equation:

X degree + x degree + 90 degree + x/2 degree = 360 degrees

We can start by simplifying the equation:

X + x + 90 + x/2 = 360

Combining like terms:

3/2x + X + 90 = 360

Next, let's isolate the terms involving x on one side of the equation:

3/2x + x = 360 - 90

Simplifying:

5/2x = 270

To solve for x, we need to multiply both sides of the equation by 2/5:

(2/5)(5/2x) = (2/5)(270)

x = 540/5

x = 108

Therefore, the value of x in the equation x degree + x degrees + 90 degrees + x/2 degree = 360 degrees is 108.

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The given expressions are algebraic equations consisting of variables and coefficients. They involve various combinations of addition and subtraction of terms.

The expressions can be simplified by combining like terms, which involves adding or subtracting coefficients that have the same variables and exponents. The simplified forms of the expressions are as follows:

   -45a³d + 75a²c - 30bc + 18bd

   -150yu + 90au - 36av + 60yv

   -90a + 105ab - 21b + 18

   150m²nz - 120m²nc + 20mn²c - 25mn²z

12) The expression 75a²c - 45a³d - 30bc + 18bd can be rearranged by combining like terms: -45a³d + 75a²c - 30bc + 18bd.

   The expression 90au - 36av - 150yu + 60yv can be rearranged by combining like terms: -150yu + 90au - 36av + 60yv.

   The expression 105ab - 90a - 21b + 18 can be rearranged by combining like terms: -90a + 105ab - 21b + 18.

   The expression 150m²nz + 20mn²c - 120m²nc - 25mn²z can be rearranged by combining like terms: 150m²nz - 120m²nc + 20mn²c - 25mn²z.

In each case, the terms with the same variables and exponents are combined by either adding or subtracting their coefficients. The simplified forms of the expressions allow for easier manipulation and analysis of the given algebraic equations.

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