Question 3(Multiple Choice Worth 5 points)
(08.02 MC)
Which of the following tables shows the correct steps to transform x2 + 6x +8=0 into the form (x-p)² =q?
[p and q are integers]

Question 3(Multiple Choice Worth 5 Points)(08.02 MC)Which Of The Following Tables Shows The Correct Steps

Answers

Answer 1

The correct steps to transform x^2 + 6x +8=0 into the form (x-p)² =q are;

x^2 + 6x +8 +1  = 0 + 1

x^2 + 6x +8 +1  = 1

x^2 + 6x + 9 = 1

(x+3)² = 1

Vertex form of a quadratic equation

The vertex form of a quadratic equation is expressed as:

a(x-h)² + k

Given the quadratic equation

x^2 + 6x +8=0

Subtract 8 from both sides

x^2 + 6x +8 - 8 = 0 - 8

x^2 + 6x  = -8

Complete the square

(x^2+6x+3^2) = -8 + 9

(x+3)² = -8 + 9

(x+3)² = 1

Hence the correct steps to transform x^2 + 6x +8=0 into the form (x-p)² =q are;

x^2 + 6x +8 +1  = 0 + 1

x^2 + 6x +8 +1  = 1

x^2 + 6x + 9 = 1

(x+3)² = 1

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Related Questions

List all numbers from the given set that are a. natural​ numbers, b. whole​ numbers, c.​ integers, d. rational​ numbers, e. irrational​ numbers, and f. real numbers.

Answers

Answer:

  See attached

Step-by-step explanation:

You want the given numbers classified as to Natural, Whole, Integer, Rational, Irrational, and/or Real.

Real

All the given numbers are real.

Irrational

The number √2 is irrational. All the others are rational.

Integer

The values √16 = 4, 0, and -5 are integer values. Whole numbers are non-negative integers, so exclude -5. Natural numbers are positive integers, so exclude 0 and -5.

The Xs indicate the categories each number belongs to.

Can someone answer this question

Answers

Answer:3rd one

Step-by-step explanation:

3rd one explanation

Given f(x) = x^2 what is the range of g(x) = f(x+2)-5

Answers

The range of the function g(x) = f(x+2)-5 is:

R = [0, ∞)

What is the range of g(x) = f(x+2)-5?

For a function y = f(x), we define the range as the set of possible outputs of the function.

Here we want to find the range of:

g(x) = f(x+2)-5

Where f(x) = x²

Replacing f(x) in the rule for g, we will get the composition:

g(x) = (x + 2)² - 5

The first part can only be a positive number or 0, then the minimum of the range is -5

And then the function only grows, then the range of g(x) is:

R = [0, ∞)

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find the area of a triangle whose base is 8 inches and whose height is 12 in.
(a) 96in.²
(b) 10in ²
(c) 20in.²
(d) 48in.²​

Answers

Answer:

48in.²

Step-by-step explanation:

The formula for the area of a triangle is 1/2 x b x h

Since the b = 8 and the h = 12, substitute 8 and 12 for b and h.

1/2 x 8 x 12

4 x 12

48in.²

Answer:

d

Step-by-step explanation:

area= ½ × b × h

= ½ × 8 × 12

= ½ × 96

= ⁹⁶/²

= 48in²

Solve and graph. |3x+6/9|>=2

Answers

The solution to the inequality expression is -8/9 ≤ x ≥ 4/9

How to solve and graph the inequality expression

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

|3x + 6/9| ≥ 2

Expand the inequality expression

So, we have

-2 ≤ 3x + 6/9 ≥ 2

Subtract 6/9 from both sides

So, we have

-24/9 ≤ 3x ≥ 12/9

Divide through the equation by 3

-8/9 ≤ x ≥ 4/9

Hence, the solution to the inequality expression is -8/9 ≤ x ≥ 4/9

The graph is attached

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Use the graph to answer the question.

Graph of polygon ABCD with vertices at negative 2 comma negative 1, 0 comma negative 4, 4 comma negative 4, 2 comma negative 1. A second polygon A prime B prime C prime D prime with vertices at 5 comma negative 1, 7 comma negative 4, 11 comma negative 4, 9 comma negative 1.

Determine the translation used to create the image.

7 units to the right
3 units to the right
7 units to the left
3 units to the left

Answers

The translation used to create the image A'B'C'D', from the pre image ABCD is T(7, 0), which corresponds with the option;

7 units to the right

What is a translation translation transformation?

The coordinates of the vertices of the polygon ABCD are;

A(-2, -1), B(0, -4), C(4, -4), D(2, -1)

The coordinates of the vertices of the polygon A'B'C'D' are;

A'(5, -1), B'(7, -4), C'(11, -4), D'(9, -1)

The coordinates of the vertices of the image indicates;

The difference between the coordinates are;

A'(5, -1) - A(-2, -1) = (5 - (-2), -1 - (-1)) = (7, 0)

B'(7, -4) - B(0, -4) = (7 - 0, -4 - (-4)) = (7, 0)

C'(11, -4) - C(4, -4) = (11 - 4, -4 - (-4)) = (7, 0)

D'(9, -1) - D(2, -1) = (9 - 2, -1 - (-1)) = (7, 0)

The translation that is used to create the image is therefore;

A translation 7 units to the right

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a librarian has weekends off and has 10 paid vacation days per year, including holidays that fall on weekends. If his salary is 32,500 per year, what is his pay per workday?

Answers

his pay per workday will be $127.45.

How to calculate interest?

Thus, the formula for calculating simple interest is J = C i t, where J is interest, C is principal, i is interest rate, and t is time. That is, J represents the amount added to the initial value, the capital is the calculation basis, the interest rate is the percentage applied on the capital and time is the capitalization period.

Knowing that:

10 paid vacation days per yearsalary is 32,500 per year

The total weekends in a year:

52 * 2 = 104 weekend days.

Total paid vacation days, including holidays that fall on weekends:

10 - 4 = 6 paid vacation days.

Total non-working days:

104 + 6 = 110 non-working days.

Total working days in a year:

365 - 110 = 255 working days.

Pay per workday = Annual salary / Total working days

= $32,500 / 255= $127.45

Therefore, the librarian's pay per workday is approximately $127.45.

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Teena uses 1/4 cup of oil for a cake. How many cakes can she make if she has 6 cups of oil?

Answers

Answer:

24 cakes.

Step-by-step explanation:

6 cups of oil divided by 1/4 cup oil per cake = 24 cakes

6/(1/4) = 24

or 6/(0.25) = 24

She can make 24 cakes with 6 cups of oil.

Find the average rate of change of f(x)=x²+2x+1 from x=4 to x=6.

Answers

The average rate of change of the function over the interval is 12

Finding the average rate of change

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

f(x) = x² + 2x + 1

The interval is given as

From x = 4 to x = 6

The function is a quadratic function

This means that it does not have a constant average rate of change

So, we have

f(4) = 4² + 2(4) + 1 = 25

f(6) = 6² + 2(6) + 1 = 49

Next, we have

Rate = (49 - 25)/(6 - 4)

Evaluate

Rate = 12

Hence, the rate is 12

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Question 5
TABLE 4 below shows the distribution of revenue among the different spheres of
government in South Africa from the 2017/2018 to 2021/2022 financial year. Some values
have been omitted.
TABLE 4: Distribution of Revenue among the different government spheres for
the financial years 2017/2018 to 2021/2022.
R(billion)
2019/20
546.1
471,4
Government
BHOJDIN
National
Provincial
Local
TOTAL
2017/18
453.4
410.6
**
2018/19
490,00
439,5
87.6
1 017,1
98.3
1:15.8
2020/21
557,5
500,4
103.3
1 161.2
2021/22
1 240.5
(adapted from DBE 2014 MLQP)
Use the table above to answer questions that follows
When the revenue of the local government sector was compared to the provincial sec
2017/18, it was found to be 20, 12% of the provincial sector.
Calculate
the missing value E, the total revenue for the period 2017/18.
ont sector always receives a larger share than t

Answers

The missing value E, the total revenue for the period 2017/18, is approximately 82.61 billion.

How to calculate the value

Local government sector's revenue in 2017/18 (L): Missing value (to be calculated)

Provincial sector's revenue in 2017/18 (P): 410.6 (billion)

According to the information provided, L is 20.12% of P. Mathematically, we can write this as:

L = 20.12/100 * P

Substituting the known value of P, we get:

L = 20.12/100 * 410.6

L ≈ 82.61 (rounded to two decimal places)

Therefore, the missing value E, the total revenue for the period 2017/18, is approximately 82.61 billion.

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Can someone give me the answer

Answers

Between 40 and 60 minutes the target heart rate strictly increasing then strictly decreasing. Therefore, option B is the correct answer.

From the given graph, in the first curve between the interval 0-10 speed is increasing,  between the interval 10-30 speed is constant and  between the interval 30-40 speed is decreasing.

In the second curve between the interval 40-50 speed is increasing,  between the interval 50-60 speed is decreasing, between the interval 60-65 speed is increasing and between the interval 65-73 speed is constant.

In the third curve between the interval 73-85 speed is increasing,  between the interval 85-92 speed is constant and  between the interval 92-100 speed is decreasing.

Therefore, option B is the correct answer.

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Mount Saint Helens, a volcano, erupted on May 18, 1980. Before eruption, Mount St. Helens was 2.95 kilometers high. use the bar diagram to find the difference in height of mount st. helens before and after the eruptions in meters PLEASE I NEED HELP :(((

Answers

The difference in height is 400 metres.

What is the difference in height of Mount St. Helens?

Height refers to vertical distance from the top to the object's base. Occasionally, it is also labeled as an altitude which measures from down to top of a surface.

To get difference in height, we will subtract the height after the eruption from the height before the eruption:

Given:

Height before eruption = 2.95 kilometers

Height after eruption = 2.55 kilometers

Difference in height = Height before eruption - Height after eruption

Difference in height = 2.95 km - 2.55 km

Difference in height = 0.4 kilometers

0.4 kilometers to metres will be:

= 0.4 * 1000 metres

= 400 metres.

Full question:

Mount St. Helens, located in Washington, erupted on May 18, 1980. Before the eruption, the volcano was 2.95 kilometers high. After the eruption, the volcano was 2.55 kilometers high. Find the difference in height of Mount St. Helens before and after the eruption.

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how many triangles?(urgent)

Answers

There are 75 triangles that can be formed using these 9 vertices.

Here we have,

We have 9 vertices arranged in a pattern where 5 vertices are above and 4 vertices are below in a parallel direction.

And we want to know how many triangles can be formed using these vertices.

In order to this,

Count the number of ways to choose 3 vertices from the 9 vertices.

This can be done using the combination formula, which is:

[tex]^{9}C_{3}[/tex] = 9! / (3! * (9-3)!)

      = 84.

Since,

A degenerate triangle is one where all three vertices lie on the same line. To count the number of degenerate triangles,

We need to count how many ways we can choose 3 vertices that lie on the same line.

There are 5 lines with 3 points each (the 5 lines with the upper vertices). So the number of degenerate triangles is:

5 [tex]^{3}C_{3}[/tex] + 4  [tex]^{3}C_{3}[/tex] = 9.

Subtract the number of degenerate triangles from the total number of triangles to get the number of non-degenerate triangles.

Therefore, the number of non-degenerate triangles is:

84 - 9 = 75.

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Compare:

58,565 ____ 58,566

Answers

Answer:

58,565 < 58,566

Step-by-step explanation:

You have to determine which number is greater. I usually start comparing from the farthest left digit (In this case, the ten thousands place.) and work my way up


hope this helped :)

What are the zeros of the function

Answers

Answer: i think c

Step-by-step explanation:


A rectangle has 2 sides that are each 6 centimeters lo[ng. The perimeter is 22 centimeters. How long are the other sides?

Answers

5 each
If you want me to show my work or explain let me know!! :)) have a great day!

Answer:

The other sides = 5 centimeters.

Step-by-step explanation:

Let's assume the length of the other two sides of the rectangle is "x" centimeters.

A rectangle has two pairs of equal sides. Therefore, we can set up the equation for the perimeter of the rectangle:

Perimeter = 2(length + width)

Given:

Length = 6 centimeters

Perimeter = 22 centimeters

Using the equation, we can substitute the given values:

22 = 2(6 + x)

Simplifying further:

22 = 12 + 2x

Subtracting 12 from both sides:

10 = 2x

Dividing both sides by 2:

5 = x

Therefore, the length of the other two sides of the rectangle is 5 centimeters.

how do you find out the area of a tent, floor included

Answers

To find the area of a tent, including the floor, we need to measure or determine the dimensions of both the tent's floor and any additional areas such as vestibules or extensions.

Floor Area: Measure the length and width of the tent's floor in the same unit of measurement (e.g., feet or meters). Multiply the length by the width to calculate the floor area. For example, if the length is 8 feet and the width is 6 feet, the floor area would be 8 feet * 6 feet = 48 square feet.Additional Areas: If the tent has vestibules, extensions, or any other separate areas, measure each area's dimensions and calculate their individual areas separately.Total Area: Once you have calculated the individual areas of the floor and any additional areas, simply add them together to find the total area of the tent, including the floor. For example, if the floor area is 48 square feet and there is an additional vestibule area of 10 square feet, the total area would be 48 square feet + 10 square feet = 58 square feet.

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pls solve this question its a nest pyq

Answers

After considering the given interval we reach the conclusion that the coefficient of t² is 1/2(3w² - 1), under the condition that the given expression is [tex](1-2 t w+t^{2})^{-1/2}[/tex] with range of (t<<1)

To evaluate the value for the given expression we have to apply the principles of binomial theorem

Then

[tex](1-2 t w+t^{2})^{-1/2} = (1 + (-2 t w + t^{2})/2 + (-2 t w + t^{2})^{2/8} + (-2 t w + t^{2})^{3/16} + ...)[/tex]

= 1 - t w + 3/8 * t² * w² - 5/16 * t³ * w³ +

The coefficient of t is the coefficient of the first term with a power of t.

Therefore, the coefficient of t is 1/2(3w² - 1).

The binomial theorem refers to the statement regarding any positive integer n, the nth power of the sum of two numbers a and b could be expressed as the sum of n + 1 terms of the form. The binomial theorem is applied in algebra and probability theory.

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Line r
goes through points (−5,2)
and (−3,8).

Line s
goes through points (−6,4)
and (2,12).
Which statement is true about lines r
and s?
Responses

Line r
is steeper than line s.
Line r is steeper than line

Line s
has a negative slope.
Line s has a negative slope.

Line s
is steeper than line r.
Line s is steeper than line

Line r
has a negative slope.

Answers

Answer: To determine which statement is true about lines r and s, we need to compare their slopes.

The slope of a line can be calculated using the formula:

slope = (change in y-coordinates) / (change in x-coordinates)

For line r, using the points (-5, 2) and (-3, 8):

slope_r = (8 - 2) / (-3 - (-5))

= 6 / 2

= 3

For line s, using the points (-6, 4) and (2, 12):

slope_s = (12 - 4) / (2 - (-6))

= 8 / 8

= 1

Now, let's analyze the statements:

Line r is steeper than line s. (False)

Since the slope of line r (3) is greater than the slope of line s (1), this statement is true.

Line r is steeper than line (Incomplete statement)

This statement is incomplete.

Line s has a negative slope. (False)

The slope of line s (1) is positive, not negative. So, this statement is false.

Line s is steeper than line r. (False)

Since the slope of line s (1) is less than the slope of line r (3), this statement is false.

Line r has a negative slope. (False)

The slope of line r (3) is positive, not negative. So, this statement is false.

Based on the analysis, the true statement about lines r and s is:

Line r is steeper than line s.

SOMEBODY PLEASE PLEASE HELP ME
ILL GIVE YOU 5 STARS, BRAINLIEST AND GIVE A HEART PLEASE HELP
(Please answer the questions and )

Answers

Part A ;

For Bus:

mean = 17.5

median = 15

mode= 15 and 16

Range= 17

For walking:

mean = 20.5

median = 20.5

mode= 20 and 21

Range= 3

How to calculate the various variables given above?

For Bus

To calculate the mean:

= 16+14+15+14+31+15/6

= 105/6

= 17.5

To calculate the median;

= 14,14,15,15,16,31

= 15+15/2 = 15

To calculate the range:

= 31-14 = 17

For walking;

To calculate the mean:

= 19+20+20+21+21+22/6

= 123/6

= 20.5 mins

To calculate the median

= 20+21/2

= 20.5

To calculate the mode:

=20 and 21

To calculate the range:

= 22-19

= 3

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Question Find the distance between the two points. (12, 5), (−12, −2)

Answers

Answer:

Distance between points is 25 Units.

Step-by-step explanation:

To find the distance between two points, we can use the distance formula. The formula is:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Given the two points: (12, 5) and (-12, -2), we can assign the values as follows:

(x1, y1) = (12, 5)

(x2, y2) = (-12, -2)

Plugging the values into the distance formula, we get:

d = sqrt((-12 - 12)^2 + (-2 - 5)^2)

= sqrt((-24)^2 + (-7)^2)

= sqrt(576 + 49)

= sqrt(625)

= 25

Therefore, the distance between the points (12, 5) and (-12, -2) is 25 units.

Using the image below, answer the following question: you are asked to pick 2
marbles out of the bag, what is the probability of picking a blue marble and then a
green marble, without replacing the blue one?
Please

Answers

The probability of picking a blue marble and then a green marble without replacing the blue one is 2/15.

From the figure we can write

Number of Blue marbles = 2

Number of Green marbles = 6

Number of Purple marbles = 2

Total number of marbles = 2 + 6 + 2 = 10

Now, Probability of picking a blue marble

= Number of Blue marbles / Total number of marbles

= 2 / 10

= 1/5

After removing the blue marble, the total number of marbles becomes 9

Probability of picking a green marble

= 6 / 9

= 2/3

So, the probability of picking a blue marble and then a green marble

Probability = (1/5) x (2/3) = 2/15.

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100 Points! Algebra question. Photo attached. Please show as much work as possible. Thank you!

Answers

The probability of the digestive tract disease, given a positive test result, is 45%.

A. Two-Way Frequency Table:

Let's create a two-way frequency table to represent the situation:

                                      Test Positive             Test Negative           Total

Disease Present                   A                                 B              5,000

Disease Not Present           C                                 D              95,000

Total                                   5,000                     95,000     100,000

In this table:

A represents the number of sheep that have the disease and test positive.B represents the number of sheep that have the disease but test negative.C represents the number of sheep that do not have the disease but test positive.D represents the number of sheep that do not have the disease and test negative.

B. Probability of Disease Given Positive Test Result:

P(Disease | Positive)

= (P(Positive | Disease) x P(Disease)) / P(Positive)

= 0.94  x 0.05 / P(Positive)

To calculate P(Positive), we can use the law of total probability:

P(Positive) = P(Positive | Disease) x P(Disease) + P(Positive | No Disease) x P(No Disease)

So, P(Positive | No Disease) = 1 - P(Test Negative | No Disease)

= 1 - 0.94 = 0.06

P(No Disease) = 1 - P(Disease) = 1 - 0.05 = 0.95

Now, P(Positive) = (0.94 x 0.05) + (0.06 x 0.95)

= 0.047 + 0.057

= 0.104

Finally, we can calculate P(Disease | Positive):

P(Disease | Positive) = (0.94 x 0.05) / 0.104

= 0.047 / 0.104

≈ 0.452

Therefore, the probability of the digestive tract disease, given a positive test result, is 45%.

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Find the sum of the finite series:
10
Σ2· (3)n-1
n=1

Answers

Answer: 59,048

Step-by-step explanation:

To find the sum of the given finite series:

Σ2· (3)n-1

n=1

where n ranges from 1 to 10, we can use the formula for the sum of a geometric series.

The formula for the sum of a geometric series is:

S = a * (r^n - 1) / (r - 1)

In this case, a = 2 (the first term of the series) and r = 3 (the common ratio).

Using the formula, we can calculate the sum:

S = 2 * (3^10 - 1) / (3 - 1)

= 2 * (59049 - 1) / 2

= (118098 - 2) / 2

= 118096 / 2

= 59048

Therefore, the sum of the given finite series is 59,048.

What are the side lengths of triangle Graph shows a triangle plotted on a coordinate plane. The triangle is at A(minus 7, 3), B(minus 3, 6), and C(5, 0). Type the correct answer in each box. If necessary, round any decimal to the nearest tenth. units units units

Answers

With the coordinate provided, the lengths of the triangle are Side AB =  5 units, Side BC = 10 units and Side CA = 12.4 units.

How do we find the sides of triangles using coordinates?

To find the side lengths of the triangle using the coordinates provided, we use the distance formul.

The distance between two points (x1, y1) and (x2, y2) is given by √((x2-x1)² + (y2-y1)²).

Side AB: Distance between A(-7, 3) and B(-3, 6)

= √((-3 - -7)² + (6 - 3)²)

= √((4)² + (3)²)

= √(16 + 9)

= √25

= 5 units

Side BC: Distance between B(-3, 6) and C(5, 0)

= √((5 - -3)² + (0 - 6)²)

= √((8)² + (-6)²)

= √(64 + 36)

= √100

= 10 units

Side CA: Distance between C(5, 0) and A(-7, 3)

= √((-7 - 5)² + (3 - 0)²)

= √((-12)² + 3²)

= √(144 + 9)

= √153

= 12.4 units

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I’m am so lost please help thank you all

Answers

Answer:

Step-by-step explanation:

148 people

Can someone answer this question

Answers

Answer:

The correct answer is (d): (x - 5) is a factor of f(x).

Step-by-step explanation:

PLEASE HELP

Given f(x)=4x^2 - 1 and g (x) = 2x- 1 , find each function
1. ( f+g )( x )
2. ( fg ) ( x )
3. ( f^n ) ( x )

Answers

The values of the composite functions are

(1) (f + g)(x) = 4x² + 2x - 2

(2) (fg)(x) = 8x³ - 4x² - 2x + 1

(3) (fⁿ)(x) = (4x² - 1)ⁿ

How to evaluate the composite functions

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

f(x) = 4x² - 1

g(x) = 2x - 1

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

(f + g)(x) = f(x) + g(x)

(f + g)(x) = 4x² - 1 + 2x - 1

Evaluate the like terms

(f + g)(x) = 4x² + 2x - 2

(fg)(x) = f(x) * g(x)

(fg)(x) = (4x² - 1) * (2x - 1)

Evaluate the products

(fg)(x) = 8x³ - 4x² - 2x + 1

(fⁿ)(x) = (f(x))ⁿ

So, we have

(fⁿ)(x) = (4x² - 1)ⁿ

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Read the description of g g below, and then use the drop-down menus to complete an explanation of why g g is or is not a function. g g relates a student to each course the student takes in a school year.

Answers

g is a function because each student is uniquely mapped to each course the student takes in a school year.

What is a function?

In Mathematics and Geometry, a function refers to any mathematical equation which is typically used to define and represent a relationship that exists between two or more variables such as an ordered pair, points on a graph or table.

Based on the description of g, we can reasonably infer and logically deduce that the relation g represent a function because the input values are uniquely mapped to the output values.

In this context, we can conclude that the description of g represents a function because a student represent the input values that is being to each course (output value) the student takes in a school year.

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1. ¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) ∴ ¬(Q ∧ P) ∨ (T ∧ R)

2. [(T ∨ Q) → (T ∨ S)] ∨ [(T ∨ Q) → (T ∨ S)]

Answers

Given statement solution is :- Regardless of the values of T, Q, and S, the whole Valid Statements & Implications is always true.

Let's analyze the two given statements:

¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) ∴ ¬(Q ∧ P) ∨ (T ∧ R)

To prove this statement, we can use logical equivalences and deductions:

¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) (Given)

¬{[¬Q ∨ (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) (Implication equivalence)

¬{[(¬Q ∨ ¬S) ∧ (¬Q ∨ R)] ∨ ¬P} → (¬P ↔ P) (Implication equivalence)

¬{(¬Q ∨ ¬S ∨ ¬P) ∧ (¬Q ∨ R ∨ ¬P)} → (¬P ↔ P) (De Morgan's Law)

(Q ∧ S ∧ P) ∨ (¬Q ∧ P) → (¬P ↔ P) (De Morgan's Law)

At this point, the formula ¬P ↔ P is a contradiction. The left side (¬P) states that P is false, while the right side (P) states that P is true. Therefore, the whole statement is always true, regardless of the values of Q, S, and P.

Since the statement is always true, we can conclude ¬(Q ∧ P) ∨ (T ∧ R) is true as well, making the second part of the question valid.

[(T ∨ Q) → (T ∨ S)] ∨ [(T ∨ Q) → (T ∨ S)]

In this statement, we have two identical sub-statements connected by a logical OR operator.

If we assume that (T ∨ Q) → (T ∨ S) is true, then the whole statement is true, regardless of the values of T, Q, and S.

If we assume that (T ∨ Q) → (T ∨ S) is false, then we need to check the other part of the statement.

Assuming (T ∨ Q) → (T ∨ S) is false means that (T ∨ Q) is true, but (T ∨ S) is false. In this case, the second part of the statement [(T ∨ Q) → (T ∨ S)] is true.

Therefore, regardless of the values of T, Q, and S, the whole Valid Statements & Implications is always true.

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