Perform the addition or subtraction and simplify.
x
x2 + x − 2
−
9
x2 − 5x + 4

By applying synthetic division concept, it can be concluded that f(x) = 4x³ + x² + x - 9 can be expressed as f(x) = (x + 1)(4x² -3x + 4) - 13

Synthetic division is a shorthand way of dividing polynomials where we can divide the coefficients of the polynomial by omitting variables and exponents. As a result, we get the coefficient of the quotient and the remainder.

Polynomial remainder theorem states that the value of p in argument b is equal to the remainder of the polynomial division p(x) / (x - b). Specifically, p(x) is divided by x - b with a remainder of zero if, and only if, b is a root of p.

We have the following polynomial:

f(x) = 4x³ + x² + x - 9

and we have k = -1

We will divide the polynomial f(x) by (x + 1). The steps are as follows:

1. Put the coefficients in a row and multiply the outside coefficient by the divisor: 4(-1)= -4

2. Add the inside coefficient to the product from the previous step: -4 + 1 = -3

3. Multiply the result from the previous step by the divisor: -3(-1) = 3

4. Add the next coefficient to the product from the previous step: 3 + 1 = 4

5. Multiply the result from the previous step by the divisor: 4(-1) = -4

6. Add the last coefficient to the product from the previous step: -4 - 9 = -13

Now we take the first coefficient and the result of the sum steps, we get the following numbers: 4, -3, 4, -13

Then we can write the expression as follows: 4x³ + x² + x - 9 = (x + 1)(4x² - 3x + 4) - 13

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The first step is to recall the equation for an arithmetic sequence: an = a1 + (n - 1)d, where a1 is the first term, an is the nth term, and d is the common difference.

Given the sum of three consecutive terms of the sequence, we know that a3 + a2 + a1 = 6. We can also recall that the product of the terms is a1 x a2 x a3 = -42.

Now, using the two equations, we can solve for the three terms:
a3 = (6 - a2 - a1)/2,
a2 = 6 - a1 - 2a3,
a1 = 6 - a2 - 2a3

Substituting the second and third equation into the first, we get:
a3 = -14/2 = -7

Therefore, the three consecutive terms of the arithmetic sequence are: a1 = -5, a2 = 2, a3 = -7.

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The polynomial in factored form is: (x-5)(x+6)(x-(3-2i))(x-(3+2i))

Since complex roots always come in conjugate pairs, we know that if 3-2i is a root, then 3+2i must also be a root. Therefore, we can write the polynomial in factored form by multiplying together the factors (x-5), (x+6), (x-(3-2i)), and (x-(3+2i)).

Alternatively, we can write the polynomial in factored form using the formula for a polynomial with complex roots:

p(x) = (x-r1)(x-r2)(x-r3)(x-r4)

where r1, r2, r3, and r4 are the roots of the polynomial. In this case, r1 = 5, r2 = -6, r3 = 3-2i, and r4 = 3+2i. Substituting these values into the formula gives us:

p(x) = (x-5)(x+6)(x-(3-2i))(x-(3+2i))

Therefore, the polynomial in factored form is written in the following form: (x-5)(x+6)(x-(3-2i))(x-(3+2i)).

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a)z-score

b)z-score > 1.64

c)comparing the z-score to a z-table

d)a majority of the population of soft drink consumers prefer Pepsi over Coke.

To test the hypothesis that a majority of the population of soft drink consumers prefer Pepsi over Coke, we will use a significance level of α = .10.

H0: The proportion of soft drink consumers who prefer Pepsi is ≤ 0.5.
Ha: The proportion of soft drink consumers who prefer Pepsi is > 0.5.

Test statistic: z-score

Rejection Region: z-score > 1.64

We will obtain the P value by comparing the z-score to a z-table.

Conclusion: Based on the P value, if it is less than or equal to the significance level of 0.10, then we reject the null hypothesis. We can conclude that there is sufficient evidence to suggest that a majority of the population of soft drink consumers prefer Pepsi over Coke.

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Answer: cos B-sin²A = cos²A-sin-B

Step-by-step explanation:

Answer: cos B-sin²A = cos²A-sin-B

Explanation:

Using the double angle identity, we can express the left side of the equation as:

cos(A + B).cos(A-B) = cos A.cos B - sin A.sin B

Now, using the Pythagorean identity, we can rewrite the right side of the equation as:

cos²A-sin-B = cos A.cos B - sin A.sin B

Hence, we can conclude that cos(A + B).cos(A-B) = cos B-sin²A = cos²A-sin-B

Answer: Read the step by step explanation for the answer.

Step-by-step explanation:

We can prove this trigonometric identity using the product-to-sum and Pythagorean identities as follows:

Starting with the left-hand side (LHS):

cos(A + B).cos(A - B)

Using the product-to-sum identity:

= [cos A cos B - sin A sin B][cos A cos B + sin A sin B]

= cos²A cos²B - sin²A sin²B

Using the Pythagorean identity:

= cos²A(1 - sin²B) - sin²A(1 - cos²B)

= cos²A - cos²A sin²B - sin²A + sin²A cos²B

= cos²A - sin²A + sin²A cos²B - cos²A sin²B

Using the Pythagorean identity:

= cos²A - sin²A + sin²A(1 - sin²B) - cos²A sin²B

= cos²A - sin²A + sin²A - sin²A sin²B - cos²A sin²B

= cos²A - sin²A + sin²A cos²B - sin²B cos²A

= cos²A - sin²A + cos²A sin²B - sin²B cos²A

Using the Pythagorean identity:

= cos²A - sin²A + cos²A - sin²B

= 2cos²A - sin²A - sin²B

= cos²A + cos²A - sin²A - sin²B

= cos²A + sin²A cos²B - sin²B

= RHS

Therefore, we have shown that (b) cos(A + B).cos(A-B) = cos B-sin²A = cos²A-sin-B.

The solution set for the inequality 6x + 1 ≤ 4x + 1 is x ≤ -3/2 can be written in interval notation as (-∞, 0].

To solve the inequality 6x + 1 ≤ 4x + 1, we need to isolate the variable on one side of the inequality. Here are the steps to do so:

1. Subtract 4x from both sides of the inequality: 6x + 1 - 4x ≤ 4x + 1 - 4x
2. Simplify: 2x + 1 ≤ 1
3. Subtract 1 from both sides of the inequality: 2x + 1 - 1 ≤ 1 - 1
4. Simplify: 2x ≤ 0
5. Divide both sides of the inequality by 2: 2x/2 ≤ 0/2
6. Simplify: x ≤ 0

The solution set in interval notation is (-∞, 0]. This means that any value of x less than or equal to 0 will satisfy the inequality.

"

Prper question:

Solve the inequality. Write the solution set in interval notation.

6x + 1 ≤ 4x + 1

"

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The options which satisfy the given diagram are

m∠5 + m∠6 = 180°

m∠2 + m∠3 = m∠6

m∠2 + m∠3 + m∠5 = 180°

The sum of the interior angles of a triangle is 180°.

The exterior angle of a triangle is equal to the sum of its interior opposite angles.

Let the Triangle be ABC and the exterior angle be ∠D , then

∠A + ∠B + ∠C =  180°

∠A + ∠B = ∠D , ∠A and ∠B are interior opposite angles of the triangle

So, the Options that satisfied

m∠5 + m∠6 = 180° { linear pair}

m∠2 + m∠3 = m∠6 {exterior angle theorem}

m∠2 + m∠3 + m∠5 = 180° {Angle Sum Property}

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Answer:0.1

Step-by-step explanation:

0.1

The complete table is

Whose card Length Width Perimeter Area

Sanya          10 cm 8cm               36cm         80 cm²

The perimeter of a shape is the distance around its edges. For a rectangle, like Sanya's greeting card, the perimeter can be found by adding up the lengths of all four sides. In this case, we have:

Perimeter = 2 × Length + 2 × Width

Perimeter = 2 × 10 cm + 2 × 8 cm

Perimeter = 20 cm + 16 cm

Perimeter = 36 cm

So, the perimeter of Sanya's greeting card is 36 centimeters. This means that if you were to walk all the way around the edges of the card, you would cover a distance of 36 centimeters.

The area of a shape is the amount of space it takes up. For a rectangle, the area can be found by multiplying its length by its width. In this case, we have:

Area = Length × Width

Area = 10 cm × 8 cm

Area = 80 cm²

So, the area of Sanya's greeting card is 80 square centimeters. This means that if you were to cover the card completely with paint or ink, you would need 80 square centimeters of paint or ink.

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Complete Question:

Sanya made greeting cards. Complete the table for their cards:

Whose card Length Width Perimeter Area

Sanya          10 cm 8cm              

Interest rates vary between lenders and can drastically change the monthly payment.

The interest rate is the cost of borrowing money from a lender. It determines how much extra you will pay on top of the principal amount borrowed.

The interest rate is a crucial factor to consider when shopping for loans because it directly affects the total cost of the loan.

Higher interest rates mean that you will pay more over the life of the loan, resulting in a higher monthly payment.

Lower interest rates mean that you will pay less over the life of the loan, resulting in a lower monthly payment.

Thus, choosing a loan with a lower interest rate can save you a significant amount of money in the long run.

Hence, Interest rates vary between lenders and can drastically change the monthly payment.

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One way to label the tick marks on the number lines is to use whole numbers for the longer tick marks and decimals for the shorter tick marks.

For example, on the first number line, you could label the longer tick marks as 0, 1, 2, 3, 4, and 5, and the shorter tick marks as 0.1, 0.2, 0.3, etc.

This fits with the structure of the base-ten system because each whole number is ten times greater than the corresponding decimal.

Another way to label the tick marks is to use fractions for the shorter tick marks. For example, on the second number line, you could label the longer tick marks as 0, 1, 2, 3, 4, and 5, and the shorter tick marks as 1/10, 2/10, 3/10, etc.

This also fits with the structure of the base-ten system because each fraction is one-tenth of the corresponding whole number.

Both of these labeling methods are accurate and fit with the structure of the base-ten system, so either one would be an appropriate way to label the tick marks on the number lines.

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The equation that shows the volume of the rectangular prism as a product of its edge lengths is (5/3)(1/3)(2/3) = 10/27 in cube. Option 2 is the correct answer.

The quantity of space within a three-dimensional object is its volume. The fundamentals of such forms are described in our page on three-dimensional shapes. Calculating volume is probably not something you will do as frequently as calculating area in the real world.

Even so, it may still be significant. Knowing how to calculate volume will help you determine things like how much room you have for packing when moving house, how much room you need for an office, or how much jam you can put into a jar. It can also be helpful for deciphering what the media means when they discuss a dam's capacity or a river's flow.

The volume of the rectangular prims is given by the formula:

V = lwh

Substituting the values we have:

V = (5/3)(1/3)(2/3) = 10/27

Hence, the equation that shows the volume of the rectangular prism as a product of its edge lengths is (5/3)(1/3)(2/3) = 10/27 in cube. Option 2 is the correct answer.

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i. The joint distribution for the coin and dice events is given by P(C,D) = P(C|D)P(D) = P(D|C)P(C), where C is the event that the coin lands heads and D is the event that the dice lands six.

ii. The prior probabilities for the coin and dice events are P(C) = 0.5 and P(D) = 1/6, since they are both fair.

iii. The full conditional distribution for the event that the coin is heads or the dice is six is given by P(C∪D) = P(C) + P(D) - P(C∩D) = 0.5 + 1/6 - (0.5)(1/6) = 0.5833.

iv. The full conditional distribution for the event that the coin is heads and the dice is six is given by P(C∩D) = P(C|D)P(D) = P(D|C)P(C) = (0.5)(1/6) = 0.0833.

v. The distribution for the coin and dice events given the event that the coin is heads or the dice is six is given by P(C,D|C∪D) = P(C∩D|C∪D)P(C∪D) = (0.0833)(0.5833) = 0.0486.

vi. The probability of observing that the coin is heads or the dice is six is given by P(C∪D) = 0.5833.

vii. The posterior distribution for the joint probability of the coin and dice events given the event that the coin is heads or the dice is six is given by P(C,D|C∪D) = P(C∩D|C∪D)P(C∪D) = (0.0833)(0.5833) = 0.0486.

viii. The marginal distribution for the event that the coin is heads and the dice is six given we know the event heads or six has occurred is given by P(C∩D|C∪D) = P(C,D|C∪D)/P(C∪D) = 0.0486/0.5833 = 0.0833.

ix. The marginal probability that the coin is heads and the dice is six given we know the event heads or six has occurred is given by P(C∩D|C∪D) = 0.0833.

x. The probability of you winning the prize is the same as the probability that the coin is heads and the dice is six, which is given by P(C∩D) = 0.0833.

Therefore, the probability of you winning the prize is 0.0833.

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The perimeter of this rectangle is equal to 14 units.

Mathematically, the perimeter of a rectangle can be calculated by using this mathematical expression;

P = 2(L + W)

Where:

For the width, we would determine the distance between the vertices (5, 6) and (5, 1)

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Distance = √[(5 - 5)² + (1 - 6)²]

Distance = √[(0² + (-5)²]

Distance = √25

Distance = 5 units.

For the length, we have:

Distance = √[(7 - 5)² + (6 - 6)²]

Distance = √[(2² + 0²]

Distance = √4

Length = 2 units.

Perimeter of this rectangle, P = 2(5 + 2)

Perimeter of this rectangle, P = 14 units.

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Complete Question:

Rectangle ABCD is graphed in the coordinate plane. The following are the vertices of the rectangle: A(5,1). B(7,1), C(7,6), and D(5,6). What is the perimeter of rectangle ABCD?

The mass of the chemicals being propelled is 0.41 kilograms.

The correct answer is: 0.41 kilograms.

To find the mass of the chemicals being propelled, we can use the formula for momentum:

momentum = mass x velocity

Since the momentum of the cracker before it is launched is zero, and the momentum after it is launched is the sum of the momentum of the cracker and the momentum of the chemicals, we can set up an equation:

0 = (1.5 kg)(15 m/s) + (mass of chemicals)(55 m/s)

Next, we can rearrange the equation to solve for the mass of the chemicals:

(mass of chemicals)(55 m/s) = -(1.5 kg)(15 m/s)

mass of chemicals = -(1.5 kg)(15 m/s) / (55 m/s)

mass of chemicals = -22.5 kg / 55 m/s

mass of chemicals = -0.41 kg

So the mass of the chemicals being propelled is 0.41 kilograms.

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We can write P(x) as P(x) = (x - 2)(x + 3)Q(x), where Q(x) is another polynomial.

This is the factored form of P(x), and it can be used to find other roots or to simplify the polynomial equation.

The factor theorem is a useful tool in finding the factors of a polynomial equation. The theorem states that if k is a root of the polynomial equation P(x) = 0, then x - k is a factor of P(x).

In our problem, we have P(2) = 0 and P(-3) = 0. This means that 2 and -3 are both roots of the polynomial equation P(x) = 0. According to the factor theorem, this means that x - 2 and x + 3 are both factors of P(x).

Therefore, we can write P(x) as P(x) = (x - 2)(x + 3)Q(x), where Q(x) is another polynomial. This is the factored form of P(x), and it can be used to find other roots or to simplify the polynomial equation.

In conclusion, the factor theorem is a useful tool in finding the factors of a polynomial equation. In our problem, we used the factor theorem to find the factors x - 2 and x + 3 of the polynomial equation P(x) = 0.

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The value of sin A in the triangle is 7/9

The complete question is added as an attachment

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

The triangle ABC

Using the sine rule, we have

sin(A) = Opposite/Hypotenuse

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

sin(A) = 7/9

Hence, the solution is 7/9

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If Bici expects to produce more than 500 bikes, then they should follow the second plan, and if they expect to produce fewer than 500 bikes, then they should follow the first plan.

Under the first plan, the total cost to design and build a prototype bicycle is $125,000, and the combined materials and labor costs for each bike made under this plan will be $225. This means that for each bike produced, the total cost will be $125,000 + $225 = $125,225.

Under the second plan, the total cost to design and build a prototype bicycle is $100,000, and the combined materials and labor costs for each bike made under this plan will be $275. This means that for each bike produced, the total cost will be $100,000 + $275 = $100,275.

To determine which plan is more cost-effective, we need to consider the breakeven point where the costs of both plans intersect:

125, 000 + 225x = 100, 000 + 275x

50x = 25, 000

x = 500 bicylces

If Bici Bicycle Company plans to produce more than 500 bikes, then the second plan would be more cost-effective. However, if Bici plans to produce fewer than 500 bikes, then the first plan would be more cost-effective.

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The solution of the equation 1/2x² + 3/8x - 2 using completing the square method are: 1.66 or -2.41.

To solve the equation 1/2x² + 3/8x -2 using completing the square method, follow these steps:

Step 1: Move the constant term to the right-hand side

1/2x² + 3/8x = 2

Step 2: Multiply both sides by 2 to clear the fraction

x² + 3/4x = 4

Step 3: Add the square of half of the coefficient of x to both sides of the equation

x² + 3/4x + (3/8)² = 4 + (3/8)²

The left-hand side is now a perfect square, specifically (x + 3/8)².

Step 4: Simplify the right-hand side of the equation

x² + 3/4x + 9/64 = 256/64 + 9/64

x² + 3/4x + 9/64 = 265/64

Step 5: Take the square root of both sides

x + 3/8 = ±√(265)/8

Step 6: Solve for x

x = -3/8 ± √(265)/8

x = 1.66 or - 2.41

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156 customers are expected to wear a black shoe.

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.

We know that, probability of an event = Number of favorable outcomes/Total number of outcomes

As per the given data:

Customers with black shoes = 13 out of 30

Customers with brown shoes = 17 out of 30

Total customers in the mall = 360

Probability that a customer wears black shoes:

P(B) = 13/30 = 0.433

Number of customers out of 360 expected to wear black shoes:

= P(B) × 360

= 0.433 × 360

= 156 customers.

Hence, 156 customers are expected to wear a black shoe.

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The work done by the force on the object is W∈ = -2.20 x 104 Nm.

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Answer:

60 pages

Step-by-step explanation:

64 + 56 = 120

180-120 = 60 pages

Step-by-step explanation:

red gumballs=6 blue gumballs=6

yellow gumballs=6 green gumballs=6

purple gumballs=6

total number of gumballs=30

probability of a student selecting a red gumballs

no of red gumballs 6 1

=------------------------ = ----- = -----

no of total gumballs 30 5

Answer: The surface area of a cylinder is given by the formula:

S = 2πr^2 + 2πrh

where r is the radius of the base, h is the height of the cylinder, and π is the mathematical constant pi.

In this case, the radius is 28 mm and the height is 30 mm.

Substituting these values into the formula, we get:

S = 2π(28)^2 + 2π(28)(30)

Simplifying the equation:

S = 2π(28)(28 + 30)

S = 2π(28)(58)

S = 32176π

Using a calculator, we can approximate the value of S to 3 significant figures:

S ≈ 101285.744 mm^2

Therefore, the surface area of the cylinder is approximately 101286 mm^2 (rounded to 3 significant figures).

Step-by-step explanation:

In conclusion, the mapping in part B informs us that the two circles are equation congruent, i.e., that their size and shape are the same and that the mapping preserves all of their geometrical characteristics.

A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.

The mapping in part B informs us that the points on the two circles correspond exactly to one another. In other words, there is exactly one point on the bigger circle for every point on the smaller circle, and vice versa.

In conclusion, the mapping in part B informs us that the two circles are congruent, i.e., that their size and shape are the same, and that the mapping preserves all of their geometrical characteristics.

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The statement  "If G is connected with 2 or more nodes, then all nodes must have a degree bigger or equal to 1." is true.

This is because a connected graph is defined as a graph in which there is a path between any two nodes. If a node has a degree of 0, that means it is not connected to any other nodes, and therefore the graph cannot be connected. So, in order for G to be connected, all nodes must have a degree of at least 1.

In conclusion, the answer is true.

A node is a point where several elements are connected.

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Answer: 0.125 and 12.5%

Step-by-step explanation:

To convert any fraction to decimal form, we just need to divide its numerator by the denominator.

Here, the fraction is 1/8 which means we need to perform 1 ÷ 8.

This gives the answer as 0.125. So, 1/8 as a decimal is 0.125

To get its percentage form, you simply multiply the obtained decimal value by 100:

0.125 x 100 = 12.5

The measurement of the missing angle is 33 degrees

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

The triangle

On the triangle, we have the following equation

tan(?) = 9.09/14

Evaluate

tan(?) = 0.6493

Take the arc tan of both sides

? = 33 degrees

Hence, the angle is 33 degrees

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The graph of the function x^2/9 - y^2/49 = 1 is graph (d)

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

x^2/9 - y^2/49 = 1

The above equation is an hyperbola

An hyperbola that has its center at the origin is represented as

x^2/a^2 - y^2/b^2 = 1

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

a^2 = 9

b^2 = 49

Evaluate

a = ±3

b = ±7

This means that the semi-major axis is a = 3 and the semi-minor axis is b = 7.

The graph with the above feature is graph (d)

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Using expression, the following provides one explanation for why there are two ways to add up the symbols on the tablet.

A statement, at least two variables or integers, and one or more arithmetic operations make up a mathematical expression. This mathematical procedure makes it possible to multiply, divide, add, or subtract quantities.

The hieroglyphic number system of ancient Egypt employed a decimal system with hieroglyphs to denote the powers of ten. It is challenging to give a precise response, though, without being able to see the precise symbols on the tablet.

It is feasible that there are two distinct ways to add the numbers represented by the symbols on the tablet, presuming that the tablet is employing a fundamental arithmetic operation like addition and that each symbol represents a different number.

Finding two different sets of integers that, when added together, yield the same answer is one way to approach this issue.

As an illustration, if the symbols stand for 2, 3, and 5, one technique to add them up would be:

2 + 3 + 5 = 10

They might also be added up as follows:

3 + 7 = 10

In this instance, the number 2 would be represented by the sign 2, and the number 5 by the symbol 3. This could be one reason for why the symbols on the tablet can be added in two different ways.

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