A deli wraps its cylindrical containers of hot food items with plastic wrap. The containers have a diameter of 3.5 inches and a height of 3 inches. What is the minimum amount of plastic wrap needed to completely wrap 7 containers? Round your answer to the nearest tenth and approximate using π = 3.14.

769.3 in2
365.4 in2
109.9 in2
52.2 in2

The value of fog(x) is 35x - 1.

To find fog(x), we need to substitute g(x) into f(x) wherever we see x in f(x).

So,

fog(x) = f(g(x)) = f(7x + 1)

Now, substitute g(x) = 7x + 1 into f(x) = 5x - 6:

f(g(x)) = 5(7x + 1) - 6

Simplifying:

fog(x) = 35x - 1

Therefore, fog(x) = 35x - 1.

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Complete question:

f(x) = 5x-6, g(x) = 7x + 1 find fog(x).

If two pairs of corresponding angles in two triangles are of equal measure, then the third set of angles must also be equal.

Let's consider two triangles, ∆ABC and ∆DEF. If two pairs of corresponding angles are of equal measure, then we can say that:

∠A = ∠D, and

∠B = ∠E

Now, we need to prove that ∠C = ∠F.

We know that the sum of angles in a triangle is 180°. Therefore:

∠A + ∠B + ∠C = 180° (for triangle ∆ABC)

∠D + ∠E + ∠F = 180° (for triangle ∆DEF)

From the above equations, we can write:

∠C = 180° - ∠A - ∠B

∠F = 180° - ∠D - ∠E

As ∠A = ∠D and ∠B = ∠E, we can substitute the values in the above equations:

∠C = 180° - ∠A - ∠B = 180° - ∠D - ∠E = ∠F

Therefore, we have proved that if two pairs of corresponding angles in two triangles are of equal measure, then the third set of angles must also be equal.

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The length of the side included between the angles A = 30° and B = 80° is :

AC = 11.2 centimeters.

To solve this problem, we need to use the formula for the area of a triangle:
A = (1/2)bh

where A is the area, b is the length of the base, and h is the height.

Let's label the triangle ABC, where angle A = 30° and angle B = 80°. We want to find the length of the side AC, which is the base of the triangle.

First, we need to find the height of the triangle. We can use the sine function to do this:

sin(80°) = h/AC

Rearranging this equation, we get:

h = AC * sin(80°)

Now we can substitute this into the formula for the area of a triangle:

50 = (1/2) * AC * h
50 = (1/2) * AC * (AC * sin(80°))
100 = AC^2 * sin(80°)
AC = sqrt(100/sin(80°))
AC ≈ 11.2 cm

So the length of the side AC is approximately 11.2 centimeters.

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In order to advance knowledge of unit personnel on evidence-based practice, it should be addressed first to the unit managers who can provide leadership and support to their staff. Therefore, the correct option is (b) unit managers. However, involving academic experts, experienced staff, and medical librarians is also important to ensure proper utilization of the best available evidence and training of staff.

When finding evidence that shows a need to advance knowledge of unit personnel on evidence-based practice, it should be addressed first to the unit managers. Unit managers are responsible for the overall functioning of the unit and can play a crucial role in implementing evidence-based practices. They can provide leadership and support to their staff in learning and incorporating new evidence-based practices into their daily work. However, it is important to involve academic experts, experienced staff, and medical librarians in the process as well to ensure that the best available evidence is being utilized and that staff members are properly trained on evidence-based practices.

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This equation represents the sum of the production of all three products (a, b, and c) in period 1, which must not exceed the limit of 250. So, X11 + X21 + X31 ≤ 250.

Xab = the production of product a in period b

We want to indicate that the limit on production of the company's 3 products in period 1 is 250.Let's define the variables for each product's production in period 1:X11 = the production of product A in period 1

X21 = the production of product B in period 1

X31 = the production of product C in period 1We want to express the limit on the total production of these three products in period 1, which is 250.

To represent this limit, we can write the inequality:X11 + X21 + X31 ≤ 250

This inequality states that the sum of the production of product A (X11), product B (X21), and product C (X31) in period 1 should be less than or equal to 250.Therefore, option B) X11 + X21 + X31 ≤ 250 is the correct representation of the limit on production for the company's 3 products in period 1.

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The value of the integral ∫_4^0 (f(x) + g(x)) dx is -2. The values of the integrals to evaluate each part separately.

We are given that ∫_4^0 f(x) dx = 7 and ∫_2^0 f(x) dx = -5. Also, we know that ∫_4^0 g(x) dx = -1 and ∫_2^0 g(x) dx = 2. We need to compute the integral ∫_4^0 (f(x) + g(x)) dx.

First, we can rewrite the integral as the sum of two integrals:

∫_4^0 (f(x) + g(x)) dx = ∫_4^0 f(x) dx + ∫_4^0 g(x) dx

Next, we can apply the Fundamental Theorem of Calculus to each integral separately. Since we are integrating from 4 to 0, we need to use the negative sign on the second integral:

∫_4^0 f(x) dx = -∫_0^4 f(x) dx

∫_4^0 g(x) dx = -∫_0^4 g(x) dx

Substituting these into the original expression, we get:

∫_4^0 (f(x) + g(x)) dx = -∫_0^4 f(x) dx - ∫_0^4 g(x) dx

Now we can substitute the given values for the integrals:

∫_4^0 (f(x) + g(x)) dx = -(7 + (-5)) - (-1 + 2)

Simplifying, we get:

∫_4^0 (f(x) + g(x)) dx = -2

Therefore, the value of the integral ∫_4^0 (f(x) + g(x)) dx is -2.

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The correlation coefficient measures only the strength of the linear relationship between variables.

The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to +1, with a value of 0 indicating no linear correlation, a value of +1 indicating a perfect positive linear correlation, and a value of -1 indicating a perfect negative linear correlation.

Therefore, it only measures the strength of the linear relationship between the two variables and not any other type of relationship such as a non-linear relationship. The correlation coefficient is calculated by dividing the covariance between the two variables by the product of their standard deviations, as shown in the formula below:

r = Cov(X,Y) / (σX * σY)

where r is the correlation coefficient, Cov(X,Y) is the covariance between X and Y, σX is the standard deviation of X, and σY is the standard deviation of Y.

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Yes, the statement is true and it is known as De Morgan's law for set theory.

The De Morgan's law in set theory states that the complement of the union of two sets is equal to the intersection of their complements. This means that if an element is not in A and not in B, then it is not in A or B. In other words, the elements that are not in the union of A and B are those that are not in A and those that are not in B. This is the same as saying that the elements that are in the complement of the union of A and B are those that are in the intersection of the complements of A and B. Therefore, (A ∪ B)' = A' ∩ B'. This law is useful in proving certain set-theoretic identities and simplifying complex expressions involving sets.

It states that the complement of the union of two sets is equal to the intersection of their complements. In symbols, it can be written as:

(A ∪ B)' = A' ∩ B'

where A' and B' denote the complements of sets A and B, respectively, and the symbol ' denotes complementation.

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Using a 95% confidence level, we can calculate a confidence interval for the difference of the population means. The results of the test and interval will help us interpret if there is a substantial difference between the two sources and where we would choose to purchase textbooks.

a. To determine if there is a significant difference in the price of textbooks from the campus bookstore and Amazon.com, we would use a two-sample t-test for independent samples.

This test compares the means of two independent samples to determine if there is a significant difference between them. In this case, our null hypothesis would be that there is no difference in the mean price of textbooks between the two sources, while the alternative hypothesis would be that there is a difference.

b. Using a 95% confidence level, we can calculate a confidence interval for the difference of the population means. Based on the sample data provided, the mean price of textbooks at the campus bookstore was $66.56, while the mean price at Amazon.com was $76.49.

The difference between the two means is $9.93. Using a two-sample t-test with a 95% confidence level, we find that the calculated t-value is 2.64 with 18 degrees of freedom. The 95% confidence interval for the difference in population means is (1.67, 18.19). This means that we can be 95% confident that the true difference in population means falls within this range.

c. Based on our results, we can conclude that there is a significant difference in the mean price of textbooks between the campus bookstore and Amazon.com. Additionally, since the confidence interval does not include zero, we can infer that this difference is not due to chance. Overall, we can say that textbooks are generally more expensive at Amazon.com compared to the campus bookstore.

However, it is worth noting that the sample size is relatively small, so it is possible that the results may not be representative of the entire population of textbooks.

Therefore, when deciding where to purchase textbooks, other factors such as convenience, availability, and additional costs (such as shipping) should also be taken into consideration.

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Triangle [tex]PQR[/tex] is a scalene acute triangle based on the given side lengths and angles.

Based on the given measurements and angles of triangle [tex]PQR[/tex], we can determine the type of triangle it is.

First, let's analyze the side lengths. [tex]PQ[/tex] is [tex]7.96[/tex], [tex]PR[/tex] is [tex]3.96[/tex], and [tex]RQ[/tex] is [tex]8.41[/tex]. Since all three sides have different lengths, the triangle cannot be equilateral or isosceles.

Next, let's consider the angles. [tex]PQR[/tex] is [tex]28[/tex] °, [tex]QRP[/tex] is [tex]70[/tex]°, and [tex]RPQ[/tex] is [tex]82[/tex]°. None of these angles are right angles ([tex]90[/tex]°), so the triangle is not a right triangle.

Now, let's examine the remaining possibilities. Since none of the side lengths are equal, the triangle cannot be isosceles. Additionally, since the largest angle [tex]RPQ[/tex] is [tex]82[/tex]°and is greater than [tex]90[/tex]°, the triangle cannot be obtuse.

By process of elimination, we can conclude that the triangle [tex]PQR[/tex] is a scalene acute triangle. This means it has three different side lengths and all angles are less than [tex]90[/tex]°.

In summary, triangle [tex]PQR[/tex] is a scalene acute triangle based on the given side lengths and angles.

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Answer:

K=48

Step-by-step explanation:

When two letters are next to each other, it means multiply them. So 12x4=48

Without the actual data, it's not possible to calculate the estimated regression equation, coefficient of determination, F test statistic, and p-value. However, I can explain the process of finding them.

To find the estimated regression equation, we use the method of least squares to find the values of the coefficients β0, β1, and β2 that minimize the sum of squared residuals. The equation would be of the form: y = β0 + β1x + β2x^2. The coefficient of determination (R^2) measures the proportion of variation in the dependent variable (traffic flow) that is explained by the independent variable (vehicle speed) and the model. It ranges between 0 and 1, with higher values indicating a better fit. It can be calculated as the ratio of explained variance to total variance. The F test statistic is used to test the overall significance of the model by comparing the variance explained by the model to the unexplained variance. It is calculated as the ratio of explained variance to unexplained variance, adjusted for the degrees of freedom. The p-value is the probability of obtaining an F test statistic as extreme or more extreme than the observed one, assuming the null hypothesis that the model has no significant effect. It is compared to a significance level (α) to decide whether to reject or fail to reject the null hypothesis. Using a significance level of .01, we can reject the null hypothesis of no significant effect if the p-value is less than .01. To predict the traffic flow for a speed of 39 miles per hour, we plug x = 39 into the estimated regression equation and solve for y. The result would be rounded to the nearest whole number.

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The total amount at the end of 3 years is GH¢272

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

The formula of simple iinterest amount is

I = Principal  + Principal * Rate * Time

substitute the known values in the above equation, so, we have the following representation

I = 200 + 200  * 12% * 3

Evaluate

I = 272

Hence, the amount is GH¢272

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Answer:

Formula for volume of sphere: V = 4/3πr³

Formula for surface area of sphere: A = 4πr²

Therefore, there are 56 different ways for the five customers to seat themselves in the theater using combination.

This is a combination problem, since the order of the seats does not matter. We can use the formula for combinations:

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

where n is the total number of items and r is the number of items to be selected.

In this case, n = 8 (the number of seats) and r = 5 (the number of customers), so we have:

C(8,5) = 8! / (5!(8-5)!) = 56

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The linear functions in descending order of y-intercepts are

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

The four linear functions

The y-intercept is the point where the graph intersects with the x-axis

i.e. when x = 0

Using the above as a guide, we have the following y-intercept values

Graph f(x) = -2

Formula f(x) = 5

Table of values f(x) = 3

Ordered pair f(x) = -1

In descending order of y-intercepts, we have

Formula f(x) = 5

Table of values f(x) = 3

Graph f(x) = -2

Ordered pair f(x) = -1

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So the sum of the geometric series is 1/17.

To determine whether the geometric series is convergent or divergent, we need to check the absolute value of the common ratio:

|-8/9| = 8/9 < 1

Since the absolute value of the common ratio is less than 1, the series is convergent.

To find the sum of the series, we use the formula for the sum of an infinite geometric series:

sum = a / (1 - r)

where a is the first term and r is the common ratio.

In this case, a = (-8)^0/9^1 = 1/9 and r = -8/9.

Therefore,

sum = (1/9) / (1 - (-8/9)) = (1/9) / (17/9) = 1/17

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The standard deviation measures the spread of the data points around the mean, rather than the spread about a different value such as "x" as the center.

No, it is not true that the standard deviation of "s" measures spread about the mean "x" as the center.

The standard deviation is a measure of the dispersion or variability of a set of data points.

It quantifies how much the individual data points deviate from the mean.

In statistics, "s" typically represents the sample standard deviation, which measures the spread or variability of the sample data around its sample mean.

The sample standard deviation is calculated by taking the square root of the variance.

On the other hand, "x" typically represents the sample mean, which is the average of the data points in a sample.

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We are given the 95% confidence interval for the population mean of the change in AHE between 1992 and 2008 as [4.393, 9227].

This means that if we were to take multiple random samples and compute the confidence intervals for the change in AHE using the same method, approximately 95% of those intervals would contain the true population mean.

CI = X ± z* (s/√n)

where X is the sample mean, s is the sample standard deviation, n is the sample size, and z* is the critical value of the standard normal distribution for the desired confidence level.

However, we can conclude that the confidence interval has a lower bound of 4.393 and an upper bound of 9227, which means that we are 95% confident that the population mean of the change in AHE between 1992 and 2008 falls between these two values.

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The linearization predicts that near the fixed point (0, 0), the phase portrait will have a vertical line.

To determine if the origin is the only fixed point for the nonlinear ODEs in (a) - (c), we need to find the points where the derivatives are zero.

(a) x = x², y = y

To find the fixed points, we set the derivatives equal to zero:

dx/dt = x² - 0 = 0

dy/dt = y - 0 = 0

From the first equation, we can see that the only solution is x = 0. Plugging this into the second equation, we get y = 0. Therefore, the origin (0, 0) is the only fixed point.

The linearization around the origin can be found by taking the Jacobian matrix:

J = [df/dx, df/dy] = [2x, 1]

Evaluating the Jacobian at (0, 0) gives J = [0, 1].

To draw the actual phase portrait, a computer program can be used to numerically integrate the ODEs and plot the solutions. The phase portrait will show the behavior of the solutions in the phase plane. It is expected that the phase portrait will match the prediction of the linear system, which is a vertical line.

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The approximate change in the volume of the cube is A) -1.92 cm³.

Let V be the volume of the cube and s be the length of one of its sides. We want to find dV/ds when s=8 and use it to approximate the change in V when s decreases by 0.01 cm. Since the cube has all sides equal, we have V = s³. Taking the derivative with respect to s, we get dV/ds = 3s². When s=8, we have dV/ds = 3(8)² = 192.

Therefore, when s decreases by 0.01 cm, we can approximate the change in V as dV ≈ dV/ds ds = 192*(-0.01) = -1.92 cm³. The exact change in V is dV = (7.99)³ - (8)³ = -15.98799 cm³, which is close to the approximate value we obtained. So A is correct option.

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Step-by-step explanation:

Trapezoid with bases = 33.6  and 16.8  

            and height = 12 nches ( which is 1 ft)

Area = height * (average of bases)

        = 12 * ( 33.6 + 16.8)/2  = 302.4  in^2

The line integral ∫cf⋅dr, where f(x,y)=yexyi + xexyj and c is r(t)=ti−8tj, 0≤t≤1, is equal to -3e - 8.

To evaluate the line integral, we first need to parameterize the curve c. In this case, we have c(t) = ti - 8tj, 0 ≤ t ≤ 1. We can then calculate dr/dt = i - 8j, and substitute it into the formula for the line integral:

∫cf⋅dr = ∫0^1 f(c(t)) ⋅ (i - 8j) dt

Substituting the function f(x,y) into the integral and evaluating, we get:

∫cf⋅dr = ∫0^1 (t * e^(t(-8))) dt = (-3e - 8)

Therefore, the line integral ∫cf⋅dr, where f(x,y)=yexyi + xexyj and c is r(t)=ti−8tj, 0≤t≤1, is equal to -3e - 8.

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The equation that represents the relationship in the table is y = x + 2.75.

To determine the equation that represents the relationship between the amount of money Jack earns, x, and the amount of money Julie earns, y, we need to identify the pattern in the data.

Looking at the given values, we can see that as Jack's earnings increase by $1.25, Julie's earnings increase by $1.25 as well.

This indicates a constant rate of change, or slope, between the two variables.

Using the point-slope form of a linear equation, we can write the equation as:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is any point on the line.

We can choose any point from the table, such as (87.50, 90.25), to substitute for (x1, y1).

The slope, m, is the change in y over the change in x, which is:

m = (91.50 - 90.25) / (88.75 - 87.50) = 1.25 / 1.25 = 1

Substituting the values into the point-slope form, we get:

y - 90.25 = 1(x - 87.50)

Simplifying, we get:

y - 90.25 = x - 87.50

y = x + 2.75.

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The key features of a dyad are its small size, symmetry, instability, interdependence, and intensity.

A dyad refers to a social relationship between two individuals or groups. In sociology, Georg Simmel was the first to identify the unique features of a dyad. Simmel believed that a dyad has distinct characteristics that make it different from other social relationships. In this answer, we will explore the key features of a dyad according to Simmel.

The first key feature of a dyad is that it is the smallest social group, consisting of only two members. Due to the small size of the dyad, each member has a more significant impact on the relationship's outcome than in a larger social group. In a dyad, each member can have a more direct and intimate connection with the other, creating a more intense emotional bond than larger groups.

The second feature of a dyad, according to Simmel, is that the relationship between the two members is more symmetrical than in larger social groups. This symmetry means that the two members are more equal in terms of their social status, power, and influence in the relationship. This symmetry allows for a more direct exchange of ideas and feelings, which can result in a deeper and more profound relationship.

The third key feature of a dyad is that it is unstable and can easily break down. Since there are only two members in a dyad, if one person leaves or is absent, the entire relationship is dissolved. This instability means that a dyad requires more maintenance and attention than larger social groups.

The fourth feature of a dyad is that the members of a dyad are more interdependent than in larger social groups. This interdependence means that the actions of one member have a more significant impact on the other member than in larger social groups. The members of a dyad rely on each other more heavily for emotional support, validation, and affirmation, which creates a strong emotional bond.

Finally, the fifth key feature of a dyad is that it is more intense and emotional than larger social groups. The intimacy and directness of the relationship create a more profound emotional bond, making the members of a dyad more invested in the relationship's outcome. This intensity can create a sense of exclusivity and a feeling of being in a close-knit group.

In conclusion, according to Simmel, the key features of a dyad are its small size, symmetry, instability, interdependence, and intensity. Understanding these characteristics is essential to understanding the dynamics of dyadic relationships, including romantic relationships, friendships, and other close interpersonal relationships. By examining these key features, we can better understand the unique aspects of a dyadic relationship and its impact on the individuals involved.

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The set S is a unit square in the uv-plane, bounded by the lines u = 0, u = 1, v = 0, and v = 1. The transformation given by x = v, y = u(1 - v^2) maps points in the uv-plane to points in the xy-plane.

To find the image of S under this transformation, we apply the transformation to each of the vertices of S:

Therefore, the image of the unit square S under the transformation x = v, y = u(1 - v^2) is the line segment connecting (0,0) and (1,0) in the xy-plane.

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The nurse should discard the contaminated supplies and start over with new sterile supplies.

Contamination of supplies during a procedure is a serious concern as it can lead to infection. In this scenario, the nurse should immediately recognize the contamination and take appropriate action to prevent infection. The correct action is to dispose of the contaminated supplies and obtain new sterile supplies before continuing the procedure.

The nurse should ensure that all supplies are handled in a sterile manner to reduce the risk of infection. Proper hand hygiene and sterile technique should be used throughout the procedure to minimize the risk of infection. By taking these steps, the nurse can ensure the safety and well-being of the client.

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The values of the equations are;

1. x = -1, y = 2

2. x = √5 , y = 5 - 2√5

Using the substitution method of solving simultaneous equations, we have;

x+y=5

y=x+3

Substitute the value of y in equation into as y in equation 1, we have that;

x + (x + 3) = 5

expand the bracket

x + x + 3 = 5

collect the like terms

2x = -2

Make 'x' the subject

x = -1

y = 2

2x+y=5, x² + y² =10

Make 'y' subject

y = 5 - 2x

Substitute the value

x² + (5 - 2x)² = 10

x² + 25 - 4x² = 10

collect like terms

-3x² = -15

Make 'x' the subject

x² = 5

x = √5

Substitute the value

y = 5 - 2(√5)

y = 5 - 2√5

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Answer: -14

Step-by-step explanation:I did this on khan

The results of the quadratic equation x² + p · x + q = 0 with real coefficients are:

Case A: p · q

Case B: (5 · p) / (2 · q) - 8

Case C: (3 / 2) · (p² - 4 · q)

Case D: 5 · p² - 11 · q

Herein we find a quadratic equation of the form x² + p · x + q = 0, where p, q are real coefficients, and whose roots are α, β. The values of the coefficients are described by these expressions:

p = - α - β, q = α · β

Now we proceed to determine α and β in terms of p and q:

p = - α - q / α

p · α = - α² - q

α² - p · α + q = 0

α = p / 2 ± (1 / 2) · √(p² - 4 · q)

p = - q / β - β

p · β = - q - β²

β² - p · β + q = 0

β = p / 2 ± (1 / 2) · √(p² - 4 · q)

Then,

α = p / 2 + (1 / 2) · √(p² - 4 · q), β = p / 2 - (1 / 2) · √(p² - 4 · q)

Finally, we proceed to solve on each expression:

Case A

[p / 2 + (1 / 2) · √(p² - 4 · q)]² · [p / 2 - (1 / 2) · √(p² - 4 · q)] + [p / 2 + (1 / 2) · √(p² - 4 · q)] · [p / 2 - (1 / 2) · √(p² - 4 · q)]²

[p² / 4 - (1 / 4) · (p² - 4 · q)] · [p / 2 + (1 / 2) · √(p² - 4 · q)] + [p² / 4 - (1 / 4) · (p² - 4 · q)] · [p / 2 - (1 / 2) · √(p² - 4 · q)]

[p² / 4 - (1 / 4) · (p² - 4 · q)] · p

p³ / 4 - (p / 4) · (p² - 4 · q)

p³ / 4 - p³ / 4 + p · q

p · q

Case B

[p / 2 + (1 / 2) · √(p² - 4 · q)] / [p / 2 - (1 / 2) · √(p² - 4 · q)] + [p / 2 - (1 / 2) · √(p² - 4 · q)] / [p / 2 + (1 / 2) · √(p² - 4 · q)]

[[p / 2 + (1 / 2) · √(p² - 4 · q)]² + [p / 2 - (1 / 2) · √(p² - 4 · q)]²] / [p² / 4 - (1 / 4) · (p² - 4 · q)]

[p² / 4 + (1 / 2) · p · √(p² - 4 · q) + p² - 4 · q + p² / 4 - (1 / 2) · p · √(p² - 4 · q) + p² - 4 · q] / [p² / 4 - (1 / 4) · (p² - 4 · q)]

(p² / 2 + 2 · p² - 8 · q) / q

(5 · p² / 2 - 8 · q) / q

(5 · p) / (2 · q) - 8

Case C

(α - β)²

α² - 2 · α · β + β²

[p / 2 + (1 / 2) · √(p² - 4 · q)]² - 2 · [p / 2 + (1 / 2) · √(p² - 4 · q)] · [p / 2 - (1 / 2) · √(p² - 4 · q)] + [p / 2 - (1 / 2) · √(p² - 4 · q)]²

p² / 4 + (1 / 2) · p · √(p² - 4 · q) + p² - 4 · q - 2 · [p² / 4 - (1 / 4) · (p² - 4 · q)] + p² / 4 - (1 / 2) · p · √(p² - 4 · q) + p² - 4 · q

5 · p² / 2 - 8 · q - p² / 2 - (1 / 2) · (p² - 4 · q)

2 · p² - 8 · q - (1 / 2) · (p² - 4 · q)

2 · p² - 8 · q - (1 / 2) · p² + 2 · q

3 · p² / 2 - 6 · q

3 · (p² / 2 - 2 · q)

(3 / 2) · (p² - 4 · q)

Case D

(2 · α + β) · (α + 2 · β)

(2 · α + β) · α + (2 · α + β) · (2 · β)

2 · α² + α · β + 4 · α · β + 2 · β²

2 · α² + 5 · α · β + 2 · β²

2 · [p / 2 + (1 / 2) · √(p² - 4 · q)]² + 5 · [p / 2 + (1 / 2) · √(p² - 4 · q)] · [p / 2 - (1 / 2) · √(p² - 4 · q)] + 2 · [p / 2 - (1 / 2) · √(p² - 4 · q)]²

2 · [p² / 4 + (1 / 2) · p · √(p² - 4 · q) + p² - 4 · q] + 5 · [p² / 4 - (1 / 4) · (p² - 4 · q)] + 2 · [p² / 4 - (1 / 2) · p · √(p² - 4 · q) + p² - 4 · q]

p² / 2 + p · √(p² - 4 · q) + 2 · p² - 8 · q + 5 · p² / 4 - (5 / 4) · (p² - 4 · q) + p² / 2 - p · √(p² - 4 · q) + 2 · p² - 8 · q

p² + 4 · p² - 16 · q + 5 · p² / 4 - (5 / 4) · p² + 5 · q

5 · p² - 11 · q

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