Please answer correctly !!!! Will mark brainliest answer !!!!!!!!!

Please Answer Correctly !!!! Will Mark Brainliest Answer !!!!!!!!!

Answers

Answer 1

Answer:

functions f(x) and g(x) share one root in common that of x=2 and each have another different root

Step-by-step explanation:

Notice that function g(x) has for roots the values x = -1 and x = 2, (these are the crossings of the x axis that you see in the graph)

On the other hand, the function f(x) has two roots that are obtained by factoring it out:

\(f(x)=x^2+x-6\\f(x)= x^2-2x+3x-6\\f(x)=x(x-2)\,+\,3(x-2)\\f(x)=(x-2)\,(x+3)\)

Therefore it has roots at x=2 and x= -3

So functions f(x) and g(x) share one root in common that of x=2 and each have another different root


Related Questions

Suppose that 650 lb of coffee are sold when the price is $4 per pound, and 400 lb are sold at $8 per pound

Answers

The average price per pound for all the coffee sold is $5.52 per pound, when 650 lb of coffee are sold when the price is $4 per pound, and 400 lb are sold at $8 per pound.

Suppose that 650 lb of coffee are sold when the price is $4 per pound, and 400 lb are sold at $8 per pound. We have to find the average price per pound for all the coffee sold.

Average price is equal to the total cost of coffee sold divided by the total number of pounds sold. We can use the following formula:

Average price per pound = (total revenue / total pounds sold)

In this case, the total revenue is the sum of the revenue from selling 650 pounds at $4 per pound and the revenue from selling 400 pounds at $8 per pound. That is:

total revenue = (650 lb * $4/lb) + (400 lb * $8/lb)

= $2600 + $3200

= $5800

The total pounds sold is simply the sum of 650 pounds and 400 pounds, which is 1050 pounds. That is:

total pounds sold = 650 lb + 400 lb

= 1050 lb

Using the formula above, we can calculate the average price per pound:

Average price per pound = total revenue / total pounds sold= $5800 / 1050

lb= $5.52 per pound

Therefore, the average price per pound for all the coffee sold is $5.52 per pound, when 650 lb of coffee are sold when the price is $4 per pound, and 400 lb are sold at $8 per pound.

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x+6=x what does x equal

Answers

Answer:

0 = -6

Step-by-step explanation:

Subtract 6 from both sides

x+6=x

x+6-6=x-6

2. Simplify

Subtract the numbers

x=x-6

3. Subtract x from both sides

x=x-6

x-x=x-6-x

4. Simplify

0 = -6

X+6=X
x-x=-6
0=-6
There is no actual answer as one side of the equation is 0. All we get is result which is 0=-6


What is the equation of the vertical line that contains the point (-3,8)?

Answers

Answer: The answer is; x=-3

Step-by-step explanation: (The work is in the image below)

Additional note: hey! I hope that this helps you! Have a good day!

What is the equation of the vertical line that contains the point (-3,8)?
What is the equation of the vertical line that contains the point (-3,8)?

Approximately what is the correlation coefficient of the scatter plot below: .: mr. O +1 O +0.5 O -0.9 O +0.2
Previous question

Answers

Approximately the correlation coefficient of the scatter plot below: .: mr. O +1 O +0.5 O -0.9 O +0.2 is +1.

As per the given scatter plot below: .: Mr. O +1 +0.5 -0.9 +0.2, the approximate correlation coefficient can be calculated as follows:

The correlation coefficient is the measure of the strength and direction of the linear relationship between two quantitative variables. It takes values between -1 and 1. If the value is negative, it indicates that as one variable increases, the other decreases, and if the value is positive, it indicates that as one variable increases, the other also increases.

Here, we can see that the scatter plot has a roughly linear relationship with a positive slope. Therefore, the approximate correlation coefficient of the scatter plot below is +1.

The correlation coefficient is +1.

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14. A loan is made for \( \$ 4800 \) with an APR of \( 12 \% \) and payments made monthly for 24 months. What is the payment amount? What is the finance charge? (4 points). 15. Find the present value

Answers

The monthly payment amount is approximately $129.45.

To find the payment amount and finance charge for the loan, we can use the formula for calculating monthly loan payments and finance charges.

The formula to calculate the monthly loan payment amount is given by:

\[ P = \frac{{r \cdot PV}}{{1 - (1+r)^{-n}}} \]

where:

P = monthly payment amount

r = monthly interest rate (APR divided by 12 months and 100 to convert it to a decimal)

PV = present value or loan amount

n = total number of payments

Given:

Loan amount (PV) = $4800

APR = 12%

Monthly payments (n) = 24

To calculate the monthly interest rate (r), we divide the annual percentage rate (APR) by 12 and convert it to a decimal:

\[ r = \frac{{12\%}}{{12 \cdot 100}} = \frac{{0.12}}{{12}} = 0.01 \]

Substituting the values into the formula, we have:

\[ P = \frac{{0.01 \cdot 4800}}{{1 - (1+0.01)^{-24}}} \]

Calculating this equation will give us the monthly payment amount.

To calculate the finance charge, we can subtract the loan amount (PV) from the total amount paid over the loan term (P * n).

Let's calculate these values:

\[ P = \frac{{0.01 \cdot 4800}}{{1 - (1+0.01)^{-24}}} \]

\[ P = \frac{{48}}{{1 - (1+0.01)^{-24}}} \]

\[ P = \frac{{48}}{{1 - 0.62889499777}} \]

\[ P \approx \frac{{48}}{{0.37110500223}} \]

\[ P \approx 129.4532449 \]

To calculate the finance charge, we can subtract the loan amount (PV) from the total amount paid over the loan term:

Total amount paid = P * n

Total amount paid = $129.45 * 24

Total amount paid = $3106.80

Finance charge = Total amount paid - PV

Finance charge = $3106.80 - $4800

Finance charge = $-1693.20

The finance charge is approximately -$1693.20. The negative sign indicates that the borrower will be paying less than the loan amount over the loan term.

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after sampling a column of water from the surface to a depth of 3000 meters (nearly 10,000 feet), a colleague abroad an oceanographic research vessel tells you that the water column is isopycnal. what does this mean? what conditions create such a situation? what would have to happen in order to create a pycnocline?

Answers

Salinity and density decrease as temperature rises. Salinity and density, therefore, have a negative relationship with temperature. The density rises as salinity rises. As the temperature drops, ocean water becomes denser. Therefore, water becomes denser as it becomes colder. The density of seawater also rises as salinity does.

Water that is less dense floats above water that is denser. The warmer water will float on top of the colder water if there are two layers of water with the same salinity. However, there is a catch! Salinity has less of an impact on water density than temperature does. Therefore, if the layer with higher salinity is considerably warmer than the layer with lower salinity, the layer with higher salinity can actually float on top of the layer with lower salinity.

As you descend to the ocean's bottom, the temperature of the water continues to drop. So, as you descend to the ocean's bottom, the density of the water keeps rising. The deep ocean is divided into layers, with the lightest water at the top and the densest water at the bottom. The ocean's depths have horizontal circulation. In other words, water flows in strata of equal density.

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Solve for k:
17/3k - 2 = 4/3(4k + 2)

Answers

Answer:

k = 14

hope it's helpful ❤❤❤❤❤

THANK YOU.

Solve for k:17/3k - 2 = 4/3(4k + 2)

e ohio lottery has a game called pick 4 where a player pays $1 and picks a four-digit number. if the four numbers come up in the order you picked, then you win $3900. a) write the probability distribution for a player's winnings. fill in the table below. for the computer to grade this one correctly make sure that your x values are from smallest to largest.

Answers

The probability of winning $3,899 is 0.0001, which is a very small probability, but still possible.


To write the probability distribution for a player's winnings in the Pick 4 game, we need to consider all the possible outcomes and their probabilities.

There are a total of 10,000 possible four-digit numbers that can be drawn in the game. Since the player has to match the numbers in the exact order, there is only one winning combination for each four-digit number. Therefore, the probability of winning is 1/10,000.

To calculate the player's winnings, we need to subtract the $1 cost of playing from the $3,900 prize. Thus, the player's net winnings can be calculated as follows:

Net Winnings = $3,900 - $1 = $3,899

The probability distribution for the player's winnings can be summarized in the following table:

| Winnings (x) | Probability (P) |
|--------------|-----------------|
| $0           | 0.9999          |
| $3,899       | 0.0001          |

Note that the table shows the possible winnings (x) in ascending order, as requested. The probability of winning $0 is 0.9999, which means that the player is most likely to lose their $1 bet.

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Which number is equivalent to 6%

Answers

Answer:

0.06

6/100

Step-by-step explanation:

6% is equivalent to the decimal number 0.06 or the fraction 6/100.

Answer:

6/100 or 0.06

Step-by-step explanation:

hope this helps!

A group of adult males has foot lengths with a mean of 26.84 cm and a standard deviation of 1.29 cm. Use the range rule of thumb to identify the limits separating values that are significantly low or significantly high. Is the adult male foot length of 23.8 cm significantly low or significantly​ high?
Significantly low values are ___ cm or lower.
Significantly high values are ___ cm or higher.
Select the correct choice below and fill in the answer​ box(es) to complete your choice.
A) The adult male foot length of 23.8 cm is not significant because it is between __ cm and __ cm
B) The adult male foot length of 23.8 cm is significantly low because it is less than __ cm
C) The adult male foot length of 23.8 cm is significantly high because it is greater than __ cm

Answers

The option B, "The adult male foot length of 23.8 cm is significantly low because it is less than 24.26 cm" is correct.

In the given question, a group of adult males has foot lengths with a mean of 26.84 cm and a standard deviation of 1.29 cm.

We have to find the adult male foot length of 23.8 cm significantly low or significantly​ high.

According to thumb rule, a value is significantly low if it is 2 standard deviations below mean and significantly high if it is 2 standard deviations above mean

Mean = 26.84 cm

Standard deviation = 1.29 cm

Significantly low value = 26.84 - 2*1.29 = 24.26 cm

Significantly high value = 26.84 + 2*1.29 = 29.42 cm

Significantly low values are 24.26 cm or lower.

Significantly high values are 29.42 cm or higher.

The adult male foot length of 23.8 is lower than the significant low value of 24.26.

So the option B, "The adult male foot length of 23.8 cm is significantly low because it is less than 24.26 cm" is correct.

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Write an expression for y more than 2

Answers

Answer:

2 + y

Step-by-step explanation:

all you must do is add y to 2

In a circle with radius 6 and angle intercepts an arc of length 3pi find the angle in radians in simplest form

Answers

In a circle with radius 6 and angle intercepts an arc of length 3π , the angle in radians in simplest form is π/2.

In a circle, the length of an arc is proportional to the angle that it intercepts. The ratio of the arc length to the circumference of the circle is equal to the ratio of the angle in radians to 2π. Thus, we can write:

(arc length) / (circumference) = (angle) / (2π)

In this problem, we are given that the circle has a radius of 6 and that the arc length is 3π. We can use the formula for the circumference of a circle, which is C = 2πr, to find the circumference of this circle:

C = 2πr = 2π(6) = 12π

Now we can use the formula above to find the angle in radians:

(3π) / (12π) = (angle) / (2π)

Simplifying this equation, we get:

angle = (3π * 2π) / 12π = 1/2 * π

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y=9
Prove • f(n) = 5n³ + n² + ylogn is O(n¹4logn) via definition of Big-0.

Answers

Answer:

f(n) is O(n¹⁴logn) by definition of Big-O.

Step-by-step explanation:

To prove that f(n) = 5n³ + n² + ylogn is O(n¹⁴logn), we need to show that there exist positive constants c and n₀ such that:

|f(n)| ≤ c|n¹⁴logn| for all n > n₀

Let's start by finding an upper bound for f(n). We can do this by simplifying the expression and getting rid of constants and lower-order terms:

f(n) = 5n³ + n² + ylogn
≤ 5n³ + n³ + n³ (since logn ≤ n³ for all n > 1)
= 7n³

Now, we can use this upper bound to find suitable values for c and n₀:

|f(n)| ≤ 7n³ ≤ 7n¹⁴/n¹¹ (since n¹¹ ≤ n³ for all n > 1)
≤ 7n¹⁴logn/n¹¹ (since logn ≤ n⁰ for all n > 1)
= 7n³logn

So, we can choose c = 7 and n₀ = 1 as our positive constants. Then, for all n > n₀, we have:

|f(n)| ≤ 7n³logn ≤ 7n¹⁴logn

Therefore, f(n) is O(n¹⁴logn) by definition of Big-O.

The ratio of the weight of an object on Planet A to the weight of the same object on Planet B is 100 to 3. If an elephant weighs 2000 pounds on Planet​ A, find the​ elephant's weight on Planet B.

Answers

The weight of the elephant on planet B given the ratio of the weights on planet A an B is 60 pounds.

What is the weight of the elephant on Planet B?

Ratio is used to compare two or more quantities together. It shows the number of times that one quantity is contained in another quantity. In this question, the weight of the elephant on Planet B is 100/3 times that of Planet A.

In order to determine the weight of the elephant on Planet B, multiply the ratio of the weight in planet B by the weight in Planet A and divide by the ratio of weight in planet A.

\(\bold{Weight} \ \text{in} \ \bold{Planet} \ \text{B} = \sf \dfrac{(3 \times 2000)}{100} = \text{60 pounds}\)

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what i can do in math 10​

Answers

Answer:

Listen carefully in class take notes go home review, remember to listen carefully in class is very important, and you need a schedule to help you plan your time.

Step-by-step explanation:

Hope it's help

Brainliest pls

Answer:

Definitely pay attention

always ask questions

take a lot of notes

go through your notes

I recommend watching some videos about the math if you're lost

solve some problems in free time to try and understand

which degenerate conic is formed when a double cone is sliced through the apex by a plane parallel to the slant edge of the cone?

Answers

The degenerate conic which is formed when a double cone is sliced through the apex by a plane parallel to the slant edge of the cone is the "parabolic conic".

What is a double cone?

A double cone is formed by two cones that are stacked one on top of the other with their vertices touching and with a common axis. If the slicing plane is parallel to the base of the double cone, it will produce a circle. If the slicing plane is perpendicular to the axis, it will produce an ellipse.

If the slicing plane is at an angle to the base, it will produce a hyperbola.If the slicing plane is parallel to the slant height, it will form a parabola rather than a conic.

a complete cone formed by straight lines through the vertex, the straight lines being indefinitely extended in both directions.

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A rectangular piece of metal is 10 inches long and 14 inches high.
Perimeter:
Area:

Answers

Answer:

Below in bold.

Step-by-step explanation:

Area = length * height

         =  10*14 = 140 in^2.

Perimeter =  2*length + 2*height

                  = 2*14 + 2*10

                  =  48 ins.

Answer:

The area is 140in²

The perimeter is 48in

Step-by-step explanation:

Hello!

The length and width of the rectangle is given as 10in and 14in.

Area

The formula for the area of a rectangle is \(\text{length x width}\)

Solve:

\(\text{length x width}\)\(\text{10in x 14in}\)\(140\text{in}^2\)

The area is 140in²

Perimeter

The formula for the perimeter of a rectangle is \(2(\text{length + width})\)

Solve:

\(2(\text{length + width})\)\(2(\text{10in + 14in})\)\(2(\text{24in})\)\(\text{48in}\)

The perimeter is 48in

The zeros of a function are the values of x for which the function is equal to zero.

Enter a number in each blank to make true statements about the function g(x)=(2x−4)(x−5).

The zeros of a function are the values of x for which the function is equal to zero.Enter a number in

Answers

Answer:

g(x)=0  when x= 2 or when x= 5

The graph of g intercepts the x-axis at x= 2 and x= 5

The zeros of g are 2 and 5

Step-by-step explanation:

Joanne cannot decide which of two washing machines to buy. The selling price of each is ​$390. The first is marked down by 40​%. The second is marked down by 10​% with an additional 30​% off. Find the sale price of each washing machine. Use pencil and paper. Explain why Joanne should buy the first washing machine rather than the second if the machines are the same except for the selling price.

Answers

Answer:

Joanne should definitely purchase the first one because it is much cheaper.

Step-by-step explanation:

So, to find the cost of each washing machine, we can simply turn the percentages into decimals and multiply to get our answer. Let's start with 40%. We bring the decimal back to spaces to get: 0.40. Then, we multiply it from the $390, because 40% 'of' means multiply:

0.40 x 390 = $156.

390 - 156 = $234

Now, the next one:

0.10 x 390 = $39

390 - 39 = $351

Then, we take 351 dollars and find the 30% off difference.

0.30 x 351 = 105.30

390 - 105.3 = $284.70

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How do you identify a mixed number as a rational number?

Answers

Well if a mixed number is a whole number, and a proper fraction represented together, and an irrational # is when a number is expressed in a p/q form or in fraction form where both the numerator and the denominator part are integers. I think you can know the difference maybe

What is the mathematical equation to equal 24 when using each of the numbers 3 4 6 and 7 only once?

Answers

The equation that can be formed from the given set of numbers

1) (3 + 4) * 6 + 7 = 24

2) 3 + 4 * (6 + 7) = 24

3) 3 + 4 + 6 + 7 = 20, so (3 + 4 + 6 + 7) * 1 = 24

4) 3 * 4 * 6 * 7 = 168, so 168 / 7 = 24

Mathematical Operations :

The mathematical “operation” refers to calculating a value using operands and a math operator. The symbol of the math operator has predefined rules to be applied to the given operands or numbers.

Mathematical Equations :

An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used.

BODMAS

According to the BODMAS rule, mathematical equations containing numerous operators must be resolved in the order of BODMAS, starting from the left.

Keep in mind that the order in which you perform the operations will affect the result. Parentheses are used to indicate the order in which operations should be performed.

In the first example above, the operation inside the parentheses (3 + 4) is done first, and then that result is multiplied by 6 and added to 7.

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Determine the equation of the parabola with focus
(
2
,
5
)
(2,5) and directrix

=
18
x=18.

Answers

The equation of the parabola with focus (2,5) and directrix x=18 is (x - 18)² + (y - 5)² = (y - (5 + (18 - 2) / 2))².

A parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating

straight line of that surface.

The focus of a parabola is a fixed point on the interior of a parabola used in the formal definition of the curve.

The directrix is a straight line perpendicular to the axis of symmetry and placed symmetrically with respect to the focus.

The axis of symmetry is the line through the focus and perpendicular to the directrix.

The vertex of a parabola is the point where its axis of symmetry intersects the curve. It is the point where the parabola changes direction or "opens

up" or "opens down.

The directrix is a fixed straight line used in the definition of a

parabola. It is placed such that it is perpendicular to the axis of symmetry and at a distance from the vertex equal to the

distance between the vertex and focus. It is the line that is equidistant to the focus and every point on the curve.Here's

the solution to the given problem:

The distance between the directrix and the focus is equal to p = 16 (since the directrix is x = 18, the parabola opens to the left, so the distance is measured horizontally)

The vertex is (h,k) = ((18+2)/2,5) = (10,5)

Then we can use the following formula: (x - h)² = 4p(y - k)

Substitute the vertex and the value of p. (x - 10)² = 64(y - 5)

Expand and simplify. (x - 10)² + (y - 5)² = 64(y - 5)

The equation of the parabola is (x - 10)² + (y - 5)² = 64(y - 5).

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What is the trademark of a linear equation? WILL GIVE BRAINLIEST. Will report if answer is absurd

Answers

Answer: They can be represented in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

Step-by-step explanation:

A zoologist researched the feeding habits of lemurs. she collected data by studying the lemurs in a zoo exhibit for a one-month period and drew conclusions based on her data. what kind of statistical study did the zoologist conduct?

Answers

The  kind of statistical study the zoologist conducted was D. observational study.

What is observational study in research?

An observational study  serves as a study  whereby the researcher simply observes the subjects without interfering with it.

This is different from carrying out experiment on them or just a theoretical approach.

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CHECK THE COMPLETE QUESTION BELOW:

A zoologist researched the feeding habits of lemurs. She collected data by studying the lemurs in a zoo exhibit for a one-month period and drew conclusions based on her data. What kind of statistical study did the zoologist conduct? A. survey B. theoretical study C. experiment D. observational study

Which ASSUMPTION is specific to the repeated measures ANOVA?
a. homogeneity of variance
b. IV measured on an interval or ratio scale
c. the same participants are tested in each level.
d. spehericity

Answers

The assumption that is specific to the repeated measures ANOVA is c. the same participants are tested in each level.

This is because in a repeated measures design, the same participants are tested multiple times under different conditions, and therefore any differences observed between conditions are assumed to be due to the manipulation of the independent variable rather than individual differences between participants.

The other assumptions listed (homogeneity of variance, IV measured on an interval or ratio scale, and sphericity) are also important for conducting a repeated measures ANOVA, but they are not specific to this type of design.

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Lorena es una estudiante que utiliza una red social cada 8 días. Su amigo Luis accede cada 6 días y su hermana Alexa ingresa cada 10 días. Si ellos coincidieron en su visita a esta red social el día 24 de julio

Answers

The next time they will coincide is on November 21. when Lorena uses a social network every 8 days, Luis logs in every 6 days, and his sister and Alexa log in every 10 days.

To find the time when all three coincided, we need to find the least common multiple (LCM) of 6, 8, and 10. The LCM of 6, 8, and 10 is given as,

6 8 10 | 2

3  4  5 | 3

1   4  5 | 4

1   1  5 | 5

1   1   1

LCM = 2 × 3 × 4 × 5 = 120

if they coincided on July 24, To find the time when all three coincided we need to add 120 days to July 24 to find the next time they will coincide.  if we add 120 days to July 24 we will get the result as November 21.

Therefore, The next time they will coincide is on November 21.

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The question is,

Lorena is a student who uses a social network every 8 days. His friend Luis logs in every 6 days and his sister Alexa logs in every 10 days. If they coincided with their visit to this social network on July 24 when will they coincide next time?

Question 2 For the following matrix Then [340]
A= [-127]
[-2-44]
(Please use a comma between two numbers.)
(a) The minors M13, M23, M33= 8,-4,10
(b)The cofactors C13, C23,C33= 8,4,10 (c) The determinant det(A) = 68

Answers

For the given matrix A, the minors M13, M23, M33 are 8, -4, and 10 respectively. The cofactors C13, C23, C33 are 8, 4, and 10 respectively. The determinant det(A) is 68.

To find the minors of a matrix, we need to find the determinants of the submatrices obtained by removing the row and column corresponding to the element of interest. In this case, the minors M13, M23, and M33 correspond to the determinants of the 2x2 submatrices obtained by removing the first row and the third column, second row and third column, and third row and third column, respectively.

To find the cofactors, we multiply each minor by a positive or negative sign based on its position in the matrix. The signs alternate starting with a positive sign for the top left element. In this case, the cofactors C13, C23, and C33 correspond to the minors M13, M23, and M33 respectively.

Finally, the determinant of a 3x3 matrix can be found by using the formula det(A) = a11C11 + a12C12 + a13C13, where a11, a12, and a13 are the elements of the first row of the matrix and C11, C12, and C13 are their corresponding cofactors. In this case, the determinant det(A) is 68.

Therefore, the minors M13, M23, M33 are 8, -4, and 10 respectively. The cofactors C13, C23, C33 are 8, 4, and 10 respectively. And the determinant det(A) is 68.

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Relating a difficult text to a personal experience can help readers
their reading.

Answers

Answer:

connect to

Step-by-step explanation:

dont know why people whole summery just put the answer

Relating a difficult text to a personal experience can certainly help readers understand and engage with the text.

What is difficult text?

Difficult text can refer to any type of written material that presents a challenge for the reader in terms of comprehension, vocabulary, or structure.

We have,

Relating a difficult text to a personal experience can certainly help readers understand and engage with the text.

When we connect new information to our existing knowledge and experiences, we are more likely to remember and comprehend the new information.

Additionally, making personal connections to the text can help readers to relate to the topic on a deeper, more emotional level, which can increase their interest and motivation to learn more.

For example,

If a reader is struggling to understand a complex scientific concept, they might try to relate it to an experience they had in their own life that involved similar principles.

This could involve recalling a time when they observed or experienced the concept in action or thinking about how the concept might be relevant to their own interests or goals. By making these connections, the reader may be able to better understand and remember the concept, as well as feel more invested in learning about it.

Overall, connecting difficult texts to personal experiences can be a useful strategy for improving reading comprehension and engagement, as well as making learning more enjoyable and meaningful.

Thus,

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What is the answer for 710 1/4 - -320 1/2

Answers

Answer:

\(1030 \frac{3}{4} \)

Step-by-step explanation:

\(710 \frac{1}{4} - - 320 \frac{1}{2} \)

➡️ \(710 \frac{1}{4} + 320 \frac{1}{2} \)

➡️ \((710 + 320) + ( \frac{1}{4} + \frac{1}{2} )\)

➡️ \(1030 + \frac{3}{4} \)

➡️ \(1030 \frac{3}{4} \)

Test the exactness of ODE, if not, use an integrating factor to make exact and then find general solution: (2xy-2y^2 e^3x)dx + (x^2 - 2 ye^2x)dy = 0.

Answers

It is requred to test the exactness of the given ODE and then find its general solution. Then, if the given ODE is not exact, an integrating factor must be used to make it exact.

This given ODE is:(2xy - 2y²e^(3x))dx + (x² - 2ye^(2x))dy = 0.To verify the exactness of the given ODE, we determine whether or not ∂Q/∂x = ∂P/∂y, where P and Q are the coefficients of dx and dy respectively, as follows: P = 2xy - 2y²e^(3x) and Q = x² - 2ye^(2x).Then, we have ∂P/∂y = 2x - 4ye^(3x) and ∂Q/∂x = 2x - 4ye^(2x).Thus, since ∂Q/∂x = ∂P/∂y, the given ODE is exact.To solve the given ODE, we have to find a function F(x,y) that satisfies the equation Mdx + Ndy = 0, where M and N are the coefficients of dx and dy respectively. This is accomplished by integrating both P and Q with respect to their respective variables. We have:∫Pdx = ∫(2xy - 2y²e^(3x))dx = x²y - y²e^(3x) + g(y), where g(y) is a function of y. We differentiate both sides of this equation with respect to y, set it equal to Q, and then solve for g(y). We have:(d/dy)(x²y - y²e^(3x) + g(y)) = x² - 2ye^(2x)Thus, g'(y) = 0 and g(y) = C, where C is a constant.Substituting the value of g(y) in the equation above, we get:x²y - y²e^(3x) + C = 0, as the general solution.The given ODE is exact, so we can solve it by finding a function that satisfies the equation Mdx + Ndy = 0. After integrating both P and Q with respect to their respective variables, we find that the general solution of the given ODE is x²y - y²e^(3x) + C = 0.

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