for the quadrant in which the following point is located determine which of the functions are positive (8,-15) cot, tan, cos, csc, sec, sin

Answers

Answer 1

Based on the point (8, -15) in the fourth quadrant, the positive trigonometric functions are cos (θ) and sec (θ).

We must take into account the signs of the functions depending on the provided point (8, -15) in order to identify which trigonometric functions are positive in the fourth quadrant.

Located in the fourth quadrant:

The x-coordinate is 8 and positive.

(-15) represents the negative y-coordinate.

Based on these coordinates, we can determine the signs of the trigonometric functions as follows:

cot (θ): cot(θ) = x/y. Since x is positive and y is negative, cot(θ) is negative in the fourth quadrant.

tan (θ): tan(θ) = y/x. Since both x and y have different signs (one positive and the other negative), tan(θ) is negative in the fourth quadrant.

cos (θ): cos(θ) = x/r, where r is the hypotenuse or radius. Since x is positive and r is always positive, cos(θ) is positive in the fourth quadrant.

csc (θ): csc(θ) = r/y. Since y is negative and r is always positive, csc(θ) is negative in the fourth quadrant.

sec (θ): sec(θ) = r/x. Since x is positive and r is always positive, sec(θ) is positive in the fourth quadrant.

sin (θ): sin(θ) = y/r. Since y is negative and r is always positive, sin(θ) is negative in the fourth quadrant.

So, based on the point (8, -15) in the fourth quadrant, the positive trigonometric functions are cos (θ) and sec (θ), while the negative trigonometric functions are cot (θ), tan (θ), csc (θ), and sin (θ).

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Related Questions

1. This summer it was over 80 degrees for 25% of the days in August.
Describe the likelihood of the temperature being over 80 degrees.

Answers

______________________________

= 25% × 31 Days ÷ 100= 7.75 Days= 7.75 Days ÷ 31 Days × 100= 25%There Is A 25% Chance

______________________________

You have a $250 card to use at a sporting goods store. Each pair of sneakers is worth $80 and a pair of sacks costs an inequality that represents the possible numbers of x pairs of socks you can buy when you buy 2 pairs of sneake

Answers

The inequality that represents the the possible numbers of x pairs of socks you can buy when you buy 2 pairs of sneakers, using a system of equations, is of:

y ≤ 7.

What is a system of equations?

A system of equations is a set of equations involving multiple variables that are related, and then operations are made to solve these equations and identify the numeric value of each variable in the context of the problem from which the system was built.

For this problem, the variables are given as follows:

Variable x: number of pairs of sneakers bought.Variable y: number of pairs of socks bought.

You have a $250 card to use at a sporting goods store. Each pair of sneakers is worth $80 and a pair of socks costs $12, hence the inequality regarding the total spending is:

80x + 12y ≤ 250.

When two pairs of sneakers are bought, we have that:

x = 2.

Hence the inequality regarding how many pairs of socks can be bought is:

80(2) + 12y ≤ 250.

160 + 12y ≤ 250.

12y ≤ 90

y ≤ 90/12

y ≤ 7. (rounded down, as you can buy 7 pairs of socks but not 8).

Missing Information

The cost of a pair of socks is of $12.

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Angela has $2,200 to invest at a simple interest rate of 4.5%. If she wants to earn $297 in interest, how long does she have to invest her money?

Answers

Answer:

I will not give you an answer, but I will give you how too get the equation started.

Step-by-step explanation:

set up an equation using:

 

x= amount invested at 3%

 

7800 - x = amount invested at 12%

 

.03(x) + .12(7800 - x) = 504

 

.03x +936 - .12x = 504

 

-.09x = -432

 

x = 4800

I need help
What is 15% off of 10$

Answers

Answer: $8.50

hope this helps :)

im not the nest at math but im pretty sure that is what is is.

Step-by-step explanation:

The improvements in survival rates after a treatment are of key interest. The old treatment has a survival rate of 75%. The expected survival rate with the new treatment is 85%. Two-sided significant difference at a level of 5% is required. With a sample size of 35, what is the expected power of the test

Answers

The power of the test is low and not sufficient to detect a significant difference between the two treatments with the given sample size of 35.

To calculate the expected power of the test, we need to consider the survival rates, the significance level, and the sample size. Let's follow these steps:

Determine the proportions
Old treatment survival rate (p1) = 0.75
New treatment survival rate (p2) = 0.85

Determine the significance level
Two-sided significant difference level (α) = 0.05

Calculate the pooled proportion
Pooled proportion (p) = (p1 + p2) / 2 = (0.75 + 0.85) / 2 = 0.80

Calculate the standard error
Standard error (SE) = √(p × (1 - p) × (1/n1 + 1/n2)) = √(0.80 × (1 - 0.80) × (1/35 + 1/35)) ≈ 0.065

Calculate the test statistic (z)
z = (p2 - p1) / SE = (0.85 - 0.75) / 0.065 ≈ 1.54

Find the critical value for the two-sided significant difference at the 5% level
z_critical = 1.96 (from a standard normal distribution table)

Calculate the power of the test
In this case, since the test statistic is smaller than the critical value (1.54 < 1.96), we cannot reject the null hypothesis at the 5% significance level.
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Please help me idk this

Please help me idk this

Answers

The answer to that question is 40
ANSWER:
We will divide it into a square and a triangle.
Area of square = s*s
=8in.*8in.
=64in.
Area of triangle=1/2* base * height
=1/2*8in.*8in.
=1/2*64in.
=32in.
Total area = 64i.+32in.
= 96in.
HOPE IT HELPS!!!!
PLEASE MARK BRAINLIEST !!!!

Jamal plans to save
for a new scooter that
costs $48.00. The
sales tax rate is 5%.
How much will Jamal
need to save?

Answers

Answer:

50.40 is the answer.

Step-by-step explanation:

If x is 5, then 6x = _____

Answers

Answer:

30

Step-by-step explanation:

6(5)=30

// have a great day //

Answer:

30

Step-by-step explanation:

answer this question

answer this question

Answers

First we need to solve the inequality

\(\\ \sf\longmapsto 2x-8<4x+5\)

\(\\ \sf\longmapsto -2x=13\)

\(\\ \sf\longmapsto x=-\dfrac{13}{2}\)

\(\\ \sf\longmapsto x=-4\dfrac{3}{4},6.5\)

Jose has a project that requires him to find the sum of the multiples of three from 3 all the way to 999, including those two numbers. If he uses the small number plus large number method, what is the sum of the multiples of three from 3-999? Note that he will also need to know how many numbers has in order to use that method.

Answers

The sum Jose wants to compute is

3 + 6 + 9 + … + 993 + 996 + 999

but since each number is a multiple of 3, he can remove a factor of 3 from eac term:

3 (1 + 2 + 3 + … + 331 + 332 + 333)

Then it's clear that the sum contains 333 terms.

Let S be the value of the sum of the first 333 natural numbers:

S = 1 + 2 + 3 + … + 331 + 332 + 333

Reversing the order of terms has no effect on this value:

S = 333 + 332 + 331 + … + 3 + 2 + 1

Now notice that terms in the same position sum to 334:

1 + 333 = 334

2 + 332 = 334

3 + 331 = 334

and so on. This means that if we pair up terms in the positions and add them up, all we are doing is doubling the value of S on the left side, and adding up 333 copies of 334 on the right side:

2S = 334 + 334 + … + 334 + 334

2S = 333 × 334

Solve for S :

S = 333 × 334 / 2 = 55,611

Then the sum Jose is considering is 3 times this value,

3 + 6 + 9 + … + 993 + 996 + 999 = 166,833

Need help on this geometry problem, solving for x.

Need help on this geometry problem, solving for x.

Answers

Answer:

A.) 30 degrees

Step-by-step explanation:

Brainly pls :)

Need help on this geometry problem, solving for x.

6 (4x6) x 9 (5x9) judjsjdjdjddjdnnddnfndndndndndndndndn

Answers

Answer:

48 600

Step-by-step explanation:

6(4×6) × 9(5×9)

Solve the numbers in brackets first

6(20) × 9(45)= 120 × 405

= 4 8 600

Answer:

6(20) × 9(45)= 120 × 405

= 4 8 600

Here is a scatter plot that shows the number of assists and points for a group of hockey players. The model, represented by y=1.5x+1.2, is graphed with the scatter plot. What does the slope mean in this situation? Based on the model, how many points will a player have if he has 30 assists?

Answers

Based on the model, 46.2 points a player will have if he has 30 assists. The graphs that show the association among two variables within a data set are called scatter plots.

The graphs that show the association among two variables within a data set are called scatter plots. It displays data points either on a Cartesian system or a two-dimensional plane. The X-axis is used to represent the independent variable and attribute, while the Y-axis is used to plot the dependent variable. These diagrams or graphs are frequently used to describe these plots.

number of points for a player with 30 assists

x = 30

y = 1.5x + 1.2

= 1.5(30) + 1.2

= 45 + 1.2

= 46.2

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Increase £2400 by 9%

Answers

2,400 increased by 9% is 2,616

Consider the indefinite integral ¹ 2x+4 dx: a) This can be transformed into a basic integral by letting Ա and du = b) Performing the substitution yields the integral c) Once we integrate and substitute, the final answer in terms of x is: If the marginal revenue for ski gloves is MR = -0.9x + 35 and R(0) = 0, find the revenue function. R(x) = =

Answers

The revenue function R(x) is obtained as R(x) = (-0.9/2)x² + 35x.

a) In order to transform ¹ 2x+4 dx into a basic integral, let:

u = 2x + 4,

du = 2 dx.

Then the integral becomes:¹ 2x+4 dx = ¹ u (1/2) du

b) Performing the substitution yields the integral:

¹ u (1/2) du = (1/2) ¹ u du

c) Once we integrate and substitute, the final answer in terms of x is:

(1/2) u² + C

= (1/2) (2x + 4)² + C

= x² + 4x + 2 + C.

Therefore, the indefinite integral of 2x + 4 is x² + 4x + 2 + C.

If the marginal revenue for ski gloves is MR = -0.9x + 35 and R(0) = 0, the revenue function R(x) can be found using the following steps:

Step 1: Integrate the marginal revenue function MR(x) to get the total revenue function TR(x):

TR(x) = ∫MR(x) dx

= ∫(-0.9x + 35) dx

= (-0.9/2)x² + 35x + C

Step 2: Use the initial condition R(0) = 0 to find the constant C:

R(0) = (-0.9/2)(0)² + 35(0) + C = 0

C = 0

Therefore, the revenue function R(x) is:

R(x) = (-0.9/2)x² + 35x

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Is the following statement true or false?
The intersection of three planes can be a ray.

Answers

The statement which says "The intersection of three planes can be a ray." is; True

How to define planes in math's?

In terms of line segments, the intersection of a plane and a ray can be a line segment.

Now, for the given question which states that the intersection of three planes can be a ray. This statement is true because it meets the definition of plane intersection.

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Answer:

False

Step-by-step explanation:

Saw a one star rating on the other answer, so I put false as my answer and got it correct.

What is the slope of the line that passes through the points (4, 6) and (-16, -18)?
Write your answer in simplest form.

Answers

Answer:

7x3=45 so substitute

Step-by-step explanation:

twelve new students are standing in a line. how many different ways can the first seven students in line order themselves? provide your answer below:

Answers

I hope this can help you out some

twelve new students are standing in a line. how many different ways can the first seven students in line

the basic fee for water remains the same but the tariff consumed increase 5%calculate the amount by which the total cost, including VAT , for the consumption of 15units of water, will increase

Answers

Step-by-step explanation:

Let's assume the basic fee for water is $X per unit and the tariff for consumption is 5% increase per unit. VAT (Value Added Tax) is calculated at a rate of Y% on the total cost, including the basic fee and the tariff.

Given that the consumption is 15 units, the total cost for the consumption of 15 units of water without VAT can be calculated as:

Total cost without VAT = Basic fee + (Tariff rate * Number of units)

Tariff rate = 5% increase per unit, so it would be 1 + 5% = 1.05

Total cost without VAT = X + (1.05 * 15) = X + 15.75

Now, let's assume the VAT rate is Z%. The total cost including VAT can be calculated as:

Total cost including VAT = Total cost without VAT + (VAT rate * Total cost without VAT)

Total cost including VAT = (X + 15.75) + (Z% * (X + 15.75))

To calculate the amount by which the total cost, including VAT, for the consumption of 15 units of water will increase, we need to subtract the initial cost from the increased cost:

Increased cost = Total cost including VAT - Total cost without VAT

Increased cost = [(X + 15.75) + (Z% * (X + 15.75))] - (X + 15.75)

Simplifying, we get:

Increased cost = Z% * (X + 15.75)

So, the amount by which the total cost, including VAT, for the consumption of 15 units of water will increase depends on the value of Z% (the VAT rate) and X (the initial basic fee for water). Once we have these values, we can calculate the increased cost accordingly.

To test a second condition only after the result of the first condition is known, ____________________ If statements can be used.

Answers

To test a second condition only after the result of the first condition is known, If statements can be used. The "else if" statement can be used to test a second condition only after the result of the first condition is known.

If statements are programming language statements that enable a computer program to make a decision on whether or not to perform a certain code block. The code within the conditional statement will only execute if the conditional expression is true. If there is a need to test a second condition only after the result of the first condition is known, one can use If statements as it allows the computer program to make decisions. An If statement checks a condition, and if the condition is true, it carries out a block of code otherwise, it skips the block and proceeds to the next block of code. A code block to be executed if the condition is true can be specified with the if statement. The conditional statement in an if statement can be used to compare two values. The two values must be of the same data type in order to be compared.

The "else if" statement can be used to test a second condition only after the result of the first condition is known.

In programming, the "if" statement is used to test a condition and execute a block of code if the condition is true. However, when we want to test multiple conditions sequentially, the "else if" statement is used. It allows us to specify an additional condition to test if the previous condition(s) evaluated to false.

Here's an example of how the "if" and "else if" statements can be used:

if (condition1) {

   // code to execute if condition1 is true

} else if (condition2) {

   // code to execute if condition1 is false and condition2 is true

} else {

   // code to execute if both condition1 and condition2 are false

}

In this case, the "else if" statement allows us to test the second condition only if the first condition is false. It provides a way to handle multiple scenarios or options based on different combinations of conditions.

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if 2/5 of pizza costs $1.00 how much will the whole pizza cost

Answers

Answer:

2/5

Step-by-step explanation:

On its municipal website, the city of Tulsa states that the rate it charges per 5 CCF of residential water is $21.62. How do the residential water rates of other U.S. public utilities compare to Tulsa's rate? The data shown below ($) contains the rate per 5 CCF of residential water for 42 randomly selected U.S. cities.10.38 9.08 11.7 6.4 12.32 14.43 15.4610.02 14.4 16.08 17.5 19.08 17.88 12.7516.7 17.25 15.54 14.7 18.81 17.89 14.818.32 15.95 26.75 22.22 22.66 20.88 23.3518.95 23.6 19.16 23.65 27.7 26.95 27.0426.89 24.58 37.76 26.41 38.91 29.36 41.55(a)Formulate hypotheses that can be used to determine whether the population mean rate per 5 CCF of residential water charged by U.S. public utilities differs from the $21.62 rate charged by Tulsa. (Enter != for ≠ as needed.)H0:Ha:(b)What is the test statistic for your hypothesis test in part (a)? (Round your answer to three decimal places.)What is the p-value for your hypothesis test in part (a)? (Round your answer to four decimal places.)(c)At α = 0.05, can your null hypothesis be rejected? What is your conclusion?Do not reject H0. The mean rate per 5 CCF of residential water throughout the United States differs significantly from the rate per 5 CCF of residential water in Tulsa.Reject H0. The mean rate per 5 CCF of residential water throughout the United States does not differ significantly from the rate per 5 CCF of residential water in Tulsa.Do not reject H0. The mean rate per 5 CCF of residential water throughout the United States does not differ significantly from the rate per 5 CCF of residential water in Tulsa.Reject H0. The mean rate per 5 CCF of residential water throughout the United States differs significantly from the rate per 5 CCF of residential water in Tulsa.(d)Repeat the preceding hypothesis test using the critical value approach.State the null and alternative hypotheses. (Enter != for ≠ as needed.)H0:Ha:Find the value of the test statistic. (Round your answer to three decimal places.)State the critical values for the rejection rule. Useα = 0.05.(Round your answers to three decimal places. If the test is one-tailed, enter NONE for the unused tail.)test statistic≤test statistic≥State your conclusion.Do not reject H0. The mean rate per 5 CCF of residential water throughout the United States differs significantly from the rate per 5 CCF of residential water in Tulsa.Reject H0. The mean rate per 5 CCF of residential water throughout the United States does not differ significantly from the rate per 5 CCF of residential water in Tulsa.Do not reject H0. The mean rate per 5 CCF of residential water throughout the United States does not differ significantly from the rate per 5 CCF of residential water in Tulsa.Reject H0. The mean rate per 5 CCF of residential water throughout the United States differs significantly from the rate per 5 CCF of residential water in Tulsa.

Answers

The null hypothesis for the data will be 21.62 and the alternate hypothesis is 2.02 for the p-value for the data is 0.2253 .

The charge at which anything happens is referred to as the velocity at which it happens.

The required details for mean rate :

(a) H0: µ = 21.62

Ha: µ ≠ 21.62

(b) t = -1.231

p-value = 0.2253

(c) Stop rejecting H0 right now. No longer significantly different from the domestic water tariff in Tulsa, the suggested household water charge per five CCF for the entire USA.

(d) H0: µ = 21.62

Ha: µ ≠ 21.62

t = -1.231

check statistic ≥ 2.020

Don't dismiss H0 any longer. The suggested five CCF residential water charge for the entirety of the USA is no longer significantly different from the five CCF residential water tariff in Tulsa.

The P-value is higher at 0.05, the level of significance. The impact in this instance is negligible. The attempt to reject the null hypothesis failed.

The conclusion is that there is insufficient statistical support to determine whether other American cities have a different mortality rate than Tulsa.

The crucial values for t at this level of significance are t=2.019.

Given that the statistic t = -1.15 is inside the acceptance range in this case, the null hypothesis is not disproved.

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a sales representative calls on four hospitals in westchester county. it is immaterial what order he calls on them. how many ways can he organize his calls?

Answers

Since the sales representative calls on four hospitals in westchester county, the numbers of ways can he organize his calls is option A: 24  possible ways.

What Exactly Is a Sales representative?

Throughout every stage of the sales process, a sales representative deals directly with customers. They must determine a customer's needs, make a compelling case for pertinent items or services, and guarantee a positive overall experience.

Note that:

For the first call he makes, he has 4 choices.

For the first call he makes, he has  3 choices.

For the first call he makes, he has choices.

For the first call he makes, he has choice.

Solution:

4 x 3 x 2 x 1 = 24 possible combinations.

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See options below

A sales representative calls on four hospitals in Westchester County. It is immaterial what order he calls on them. How many ways can he organize his calls?

24

7

120  

37

How many cups are 2 pounds?

Answers

2 pounds is equivalent to approximately 9 cups. both are unit of measurement of weight.

There are several ways to convert weight measurements, such as pounds, to volume measurements, such as cups. However, it is important to note that the conversion will vary depending on the substance being measured.

In general, one pound of a dry ingredient such as flour or sugar is equivalent to approximately 2 cups. Therefore, 2 pounds is equivalent to approximately 4 cups. However, this conversion may vary depending on the substance and the measurement method.If you are measuring a liquid, 1 pound is equivalent to 16 ounces.

1 cup is equivalent to 8 ounces. Therefore 2 pounds is equivalent to 32 ounces or 4 cups.It's also important to note that weight measurements are more precise than volume measurements, as volume measurement can be affected by factors such as how tightly packed the ingredient is, so it is best to use weight measurements for accurate measurements.

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Gardial GreenLights, a manufacturer of energy-efficient lighting solutions, has had such success with its new products that it is planning to substantially expand its manufacturing capacity with a $20 million investment in new machinery. Gardial plans to maintain its current 50% debt-to-total-assets ratio for its capital structure and to maintain its dividend policy in which at the end of each year it distributes 25% of the year's net income. This year's net income was $8 million. How much external equity must Gardial seek now to expand as planned

Answers

To expand its manufacturing capacity with a $20 million investment in new machinery while maintaining its current capital structure and dividend policy, Gardial GreenLights must seek external equity of $8 million.

The capital structure of Gardial GreenLights is characterized by a 50% debt-to-total-assets ratio. Since the company plans to maintain this ratio, the remaining 50% of the assets must be financed through equity. The investment in new machinery is $20 million, which represents 50% of the total assets needed for expansion. Therefore, the total assets required for expansion would be $40 million (20 million divided by 0.5). To determine the amount of external equity needed, we subtract the company's existing equity from the total assets required. Given that the company's dividend policy entails distributing 25% of the year's net income, we can calculate the retained earnings. This year's net income was $8 million, and 25% of that ($2 million) would be distributed as dividends. Therefore, the retained earnings amount to $6 million ($8 million - $2 million). Since external equity is the difference between the total assets required and retained earnings, Gardial GreenLights must seek external equity of $34 million ($40 million - $6 million) to finance its expansion plans.

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The equation of a parabola is y=2x^2 +8x +3

Write the equation in vertex form and show your work.

Answers

Answer: y = 2(x + 2)² - 5

Step-by-step explanation:

          We are going to use the completing the square method to transform this quadratic equation from standard form to vertex form.

Given:

     y = 2x² + 8x + 3

Factor the 2 out of the first two terms:

     y = 2(x² + 4x) + 3

Add and subtract \(\frac{b}{2} ^2\):

     y = 2(x² + 4x + 4 - 4) + 3

Distribute the 2 into -4 and combine with the 3:

     y = 2(x² + 4x + 4) - 5

Factor (x² + 4x + 4):

     y = 2(x + 2)² - 5

On a recent survey, 40% of the 900 people surveyed preferred running.
How many people preferred running?

Answers

Answer: 550

Explanation:

Well first, you need to solve 40 percent of 900 (which is 360) then subtract it from the main amount (900)

So 900-360

Answer:

360

Step-by-step explanation:

So we have 40% and 900 people. Before we ask ourselves what is 40 percent of 900, lets ask ourselves what is 10 percent. So 10 percent of 900 would be 90 because you have to move the decimal point 1 unit to the left.        

Now, 40 percent is 4 times the value of 10 percent so 40 percent of 900 has to be 360, because 90 times 4 is 360.

And you can check it by multiplying 900 x .40 which would be 360

Hope this helped, good luck :)

What's the slope of the table

What's the slope of the table

Answers

Answer:

8 3 5 4 6 8 8 4 8 8. 8 8 8 8 8 8 8 7 7 7 7 77 7 4 4. 5 8

Write the equation of the line, in general form, that has a slope of 3 and a y-intercept of -5.

Answers

Answer:

y = 3x - 5

Step-by-step explanation:

To find the equation of the line, you can use y = m x + c.

Here,

m ⇒ slope

c ⇒ y-intercept

Given that,

slope = 3

y-intercept = -5

So, to find the equation of the line you have to simply replace m and c in the formula, (y = m x + c )with the given slope and y-intercept.

Let us solve it now.

y = m x + c

y = 3x - 5

Answer:

y = 3x - 5

Step-by-step explanation:

Slope-intercept form of an equation

y = mx + bm = slopeb = y-intercept

Given that m = 3 and b = -5

y = 3x - 5

Solve the given differential equation by undetermined coefficients.
y'' + 6y = −294x2e6x
y(x) =
Solve the given differential equation by undetermined coefficients y(x)Solve the given differential equation by undetermined coefficients.
y'' + 6y = −294x2e6x
y(x) =

Answers

The solution of the differential equation by undetermined coefficients y(x) is c1 cos(√6x) + c2 sin(√6x) − 49/3 x²e⁶ˣ

The differential equation given is a second-order linear homogeneous differential equation with constant coefficients, which has the form:

y'' + ay' + by = f(x)

where a and b are constants and f(x) is a non-homogeneous function. To solve this equation, we need to find a particular solution for the non-homogeneous part and add it to the general solution of the homogeneous part.

The method of undetermined coefficients is used to find the particular solution for non-homogeneous differential equations. In this method, we assume the particular solution has the same form as the non-homogeneous function, with unknown coefficients that need to be determined.

In this case, the non-homogeneous function is −294x²e⁶ˣ. We assume that the particular solution has the form:

y_p(x) = Ax²e⁶ˣ

where A is the unknown coefficient that we need to determine.

Now, we find the first and second derivatives of y_p(x):

y'_p(x) = 2Axe⁶ˣ + 6Ax²e⁶ˣ

y''_p(x) = 12Axe⁶ˣ + 12Ax²e⁶ˣ

Substituting these derivatives and the assumed particular solution into the differential equation, we get:

(12Axe⁶ˣ + 12Ax²e⁶ˣ) + 6(Ax²e⁶ˣ) = −294x²e⁶ˣ

Simplifying the equation, we get:

18Ax²e⁶ˣ = −294x²e⁶ˣ

Solving for A, we get:

A = −49/3

Therefore, the particular solution is:

y_p(x) = −49/3 x²e⁶ˣ

Now, we need to find the general solution for the homogeneous part of the differential equation:

y'' + 6y = 0

The characteristic equation is:

r² + 6 = 0

Solving for r, we get:

r = ±√6i

Therefore, the general solution for the homogeneous part is:

y_h(x) = c1 cos(√6x) + c2 sin(√6x)

where c1 and c2 are constants.

Finally, the general solution for the non-homogeneous differential equation is the sum of the particular and homogeneous solutions:

y(x) = y_h(x) + y_p(x)

y(x) = c1 cos(√6x) + c2 sin(√6x) − 49/3 x²e⁶ˣ

where c1 and c2 are determined by the initial conditions, if given.

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