The value of a in the equation X^2 - 10x + 25 = (X-a)^2 is 5
Calculating the value of a in the equationFrom the question, we have the following parameters that can be used in our computation:
X^2 - 10x + 25 = (X-a)^2
When the equation on the left hand side is factorized
We have
(X - 5)^2 = (X - a)
By comparing both sides of the equation, we have
-5 = -a
This gives
a = 5
Hence, the solution is a = 5
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Can someone help me asap? It’s due today!!
Answer:
D. There are no outliers
Step-by-step explanation:
So if you open your eyes and count the numbers you'll realize that 2 is in fact an outlier
Let X1, X2, ... , X30 be a random sample of size 30 from a Poisson distribution with a mean of 2/3. Approximate (a) P 15 < 30 i=1 Xi 22 . (b) P 21 30 i=1 Xi < 27 .
(a) To approximate P(15 < ΣXi < 22), we can use the Central Limit Theorem (CLT) since we have a large enough sample size (n = 30) and the mean and variance of the Poisson distribution are both finite.
First, we need to find the mean and variance of the sample mean, which is also the mean and variance of the Poisson distribution:
μ = λ = 2/3
σ^2 = λ = 2/3
Next, we can standardize the random variable Z = (ΣXi - nμ) / sqrt(nσ^2) to have a standard normal distribution:
Z = (ΣXi - 30(2/3)) / sqrt(30(2/3)) = (ΣXi - 20) / sqrt(20)
Then, we can use a standard normal table or calculator to find the probability:
P(15 < ΣXi < 22) ≈ P(-2.74 < Z < -1.77) ≈ 0.038
Therefore, the approximate probability is 0.038.
(b) To approximate P(21 < ΣXi < 27), we can use the same method with the CLT.
First, we need to find the mean and variance of the sample mean:
μ = λ = 2/3
σ^2 = λ = 2/3
Next, we can standardize the random variable Z = (ΣXi - nμ) / sqrt(nσ^2) to have a standard normal distribution:
Z = (ΣXi - 30(2/3)) / sqrt(30(2/3)) = (ΣXi - 20) / sqrt(20)
Then, we can use a standard normal table or calculator to find the probability:
P(21 < ΣXi < 27) ≈ P(-0.68 < Z < 0.68) ≈ 0.495
Therefore, the approximate probability is 0.495.
Are the data shown in this line plot skewed left, skewed right, or not skewed?
Responses
skewed left
not skewed
skewed right
The data plot in the line plot are skewed right
Given data ,
Let the data be shown in the graph be a line plot
Now , A line plot, also known as a dot plot, is a graphical representation of data where data points are shown as dots along a number line.
A right-skewed line plot would have more dots towards the lower end of the number line, with fewer dots towards the higher end.
From the line plot , we can see that most data points are placed on the lower end of the line plot
Hence , the data is skewed right
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Seven cousins will devide the contents of a box of popcorn among them. If the box contains 43.75 ounces of popcorn how many ounces of popcorn will each cousins get
Answer: 6.25
Step-by-step explanation:
You have to divided 43.75 ounces by 7 which equals 6.25
You select a marble without looking and then put it back. If you do this 72 times, what is the best prediction possible for the number of times you will pick a marble that is not blue?
Answer:
Step-by-step explanation: I might not be exact, cause im still a beginner, but, maybe halve of 72?
Identify the prime numbers in the following lists. a. 5, 7, 12, 11b. 14, 6, 3c. 31, 17, 9, 15d. 782, 61, 2
a. The prime numbers in this list are 5 and 7. b. There are no prime numbers in this list. c. The prime numbers in this list are 31 and 17. d. The prime numbers in this list are 61 and 2.
a. In the list 5, 7, 12, 11, the prime numbers are 5, 7, and 11. Prime numbers are numbers greater than 1 that have only two factors, 1 and themselves.
b. In the list 14, 6, 3, the prime number is 3. The other numbers have more than two factors, making them not prime.
c. In the list 31, 17, 9, 15, the prime numbers are 31 and 17. The numbers 9 and 15 have more than two factors, so they are not prime.
d. In the list 782, 61, 2, the prime numbers are 61 and 2. Number 2 is the smallest prime number, and 61 has only two factors, 1 and 61. The number 782 has more than two factors, so it is not prime.
To summarize, the prime numbers in each list are: a. 5, 7, 11; b. 3; c. 31, 17; d. 61, 2.
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The derivation of the Quadratic Formula is shown by completing the square and applying square roots. What is the missing step?
A. x^2+b/a · x+c=0
B. ax^2+b/a · x+c/a=0
C. ax^2+b/a · x+c=0
D. x^2+b/a · x+c/a=0
Answer: x^2 + bx/a + c/a = 0
Step-by-step explanation:
the step simplifies the division of a:
0 divided by anything is still zero, so you can remove the a denominator
a/a = 1, and is therefore irrelevant, so ax^2 / a = x^2
the rest can stay
T/F; A line, ray, or segment that divides a line segment into two equal parts. It also makes a right angle with the line segment.
A line, ray, or segment that divides a line segment into two equal parts. It also makes a right angle with the line segment. The given statement is false.
A perpendicular bisector is a line, ray, or segment that intersects a line segment at its midpoint and creates two equal parts. It is called "perpendicular" because it always intersects the line segment at a 90-degree angle, but it does not necessarily make a right angle with the line segment. A line, ray, or segment that divides a line segment into two equal parts is called the perpendicular bisector.
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solve for y=
8y-9=14
y= ?
Step-by-step explanation:
8y-9=14
bring the 9 to the other side because of which the operator has to change to a plus sign.
8y=14+9=23
23÷8=
[tex] \frac{23}{8} [/tex]
or
[tex]2.875[/tex]
Find the sector area
The area of the sector as shown in the diagram is 13.1 cm².
What is sector?A sector is said to be a part of a circle made of the arc of the circle along with its two radii.
To calculate the area of the sector, we use the formula below
Formula:
A = ∅πr²/360°............................ Equation 1Where:
A = Area of the sector∅ = Angle formed by the sector at the center of the circler = Radius of the circleFrom the question,
Given:
r = 5 cm∅ = π/3Note that 2π = 360°π = 3.14Substitute these values into equation 1
A = (π/3)×3.14×5²/2πA = 13.1 cm²Hence, the area of the sector is 13.1 cm².
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Graph the absolute value function f(x) = |x – 2| on the coordinate plane.
The graph of the absolute value function f(x) = |x - 2| is given by the image presented at the end of the answer.
What is a translation?A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.
The four translation rules for functions are defined as follows:
Translation left a units: f(x + a).Translation right a units: f(x - a).Translation up a units: f(x) + a.Translation down a units: f(x) - a.The parent absolute value function is given as follows:
f(x) = |x|.
Which has the format of the V-graph, with vertex at the origin.
f(x) = |x - 2| is a translation right two units of f(x) = |x|, hence the vertex will be at the point (2,0).
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Question 2 (Essay Worth 10 points)
(01.02, 01.03 MC)
Part A: Maci made $300 grooming dogs one day with her mobile dog grooming business. She charges $60 per appointment and earned $60 in tips. Write an equation
to represent this situation and solve the equation to determine how many appointments Maci had. (5 points)
Part B: Logan made a profit of $350 as a mobile groomer. He charged $55 per appointment and received $35 in tips, but also had to pay a rental fee for the truck of
$10 per appointment. Write an equation to represent this situation and solve the equation to determine how many appointments Logan had. (5 points)
Answer:
Part A: 4
Part B: 7
Step-by-step explanation:
Part A:
60x + 60 = 300
60x = 240
x = 4
Part B:
55x - 10x = 350
45x = 350
x = 7
You will construct the incenter. You will need to construct the angle bisectors of
To construct the incenter of a triangle, construct the angle bisectors of each angle of the triangle. Here are the steps to construct the incenter of a triangle:
Draw the three sides of the triangle using a straightedge.
Choose one of the angles of the triangle and draw its bisector. To do this, place the point of your compass on the vertex of the angle, and draw an arc that intersects both sides of the angle. Without changing the width of your compass, place it on each of the two points where the arc intersects the sides of the angle, and draw two more arcs that intersect. The point where these two arcs intersect is the point where the angle bisector meets the opposite side of the triangle.
Repeat step 2 for the other two angles of the triangle. You should now have three lines that intersect at a single point.
The point where all three angle bisectors intersect is the incenter of the triangle.
Note that the incenter of a triangle is the center of the circle that can be inscribed inside the triangle.
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A charity organization is having a fundraiser. P represents the fundraiser's profit (in dollars) if n tickets are sold. A negative profit means the expenses exceeded the income from tickets. P=70n-1500. What are the expenses of the fundraiser?
what is the nature of the roots? Discriminant:81
The nature of the roots of a quadratic equation in which case, the discriminant is; 81 is; The roots are 2 distinct real roots.
What is the nature of the roots given a discriminant of 81?It follows from the task content that the nature of the roots of a quadratic equation in which case, the discriminant is 81 is to be determined.
Recall; when the discriminant of a quadratic equation is greater than 0; the equation is said to have two real distinct roots.
Ultimately, for the given equation in discuss whose discriminant is 81 is such that; it has two real distinct roots.
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The location of the obstacle course, Transformation Fitness Studios, often gets crowded. Therefore, the company opened a second location, Reflections of You, on the other side of town. The obstacle course at Reflections of You is the same as the one at Transformation Fitness Studios, except it is a horizontal reflection. This means all the checkpoints have been reflected in the y-
axis.
Using the checkpoints at Transformation Fitness Studios, A(0,5),
B(−9,5),
and C(−4,5),
find the coordinate locations of their respective checkpoints, A′,
B′,
and C′,
at the Reflections of You obstacle course.
The requried coordinate locations of the respective checkpoints at the Reflections of You obstacle course are A' (0,5), B' (9,5), and C' (4,5).
To find the reflected coordinates of each checkpoint in the y-axis, we need to negate the x-coordinate while keeping the y-coordinate the same.
For checkpoint A(0,5), the reflected coordinates A' will be (-0,5) or just (0,5) since reflecting a point over the y-axis does not change its y-coordinate.
For checkpoint B(-9,5), the reflected coordinates B' will be (9,5) since negating -9 gives us 9 and the y-coordinate remains the same.
For checkpoint C(-4,5), the reflected coordinates C' will be (4,5) since negating -4 gives us 4 and the y-coordinate remains the same.
Therefore, the coordinate locations of the respective checkpoints at the Reflections of You obstacle course are A' (0,5), B' (9,5), and C' (4,5).
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In the year 2000, the average car had a fuel economy of 21.17 MPG. You are curious as to whether this average is greater than today. The hypotheses for this scenario are as follows: Null Hypothesis: μ ≤ 21.17, Alternative Hypothesis: μ > 21.17. You perform a one sample mean hypothesis test on a random sample of data and observe a p-value of 0.0427. What is the appropriate conclusion? Conclude at the 5% level of significance.
This suggests that there is enough evidence to support the alternative hypothesis, indicating that the average fuel economy today is greater than 21.17 MPG.
Based on the given information, the null hypothesis (H₀) states that the average fuel economy is less than or equal to 21.17 MPG (μ ≤ 21.17), while the alternative hypothesis (H₁) states that the average fuel economy is greater than 21.17 MPG (μ > 21.17). You conducted a one-sample mean hypothesis test and observed a p-value of 0.0427.
To conclude at the 5% level of significance (α = 0.05), you compare the p-value to the significance level. Since the p-value (0.0427) is less than the significance level (0.05), you reject the null hypothesis.
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After selling some old books and toys, Gwen and her brother Max had 5 one-dollar bills, 6 quarters, and 8 dimes.
Part A:
What is the total amount of money earned?
Part B:
Max and Gwen decide to divide the money equally. Max said that he and Gwen cannot get even amounts of money because 5 one-dollar bills cannot be divided evenly. Do you agree with Max? Explain.
a) The total amount of money earned is 7.30 dollars.
b) Max is wrong, it is possible to divide the money equally between Gwen and Max, even though the total amount is an odd number.
a) To find the total amount of money earned, we need to convert the number of quarters and dimes to dollars.
6 quarters = 6 * 0.25 = 1.50 dollars
8 dimes = 8 * 0.10 = 0.80 dollars
The total amount of money earned is:
5 dollars + 1.50 dollars + 0.80 dollars = 7.30 dollars
b) Max is incorrect. The total amount of money earned is an odd number ($7.30), which means it is impossible to divide it equally between two people in whole dollar amounts. However, it is possible to divide the total amount equally if we allow for fractions of a dollar.
For example, if we split the money equally, Gwen and Max would each get $3.65. This means that Gwen would get 3 dollars and 65 cents, while Max would also get 3 dollars and 65 cents.
Therefore, it is possible to divide the money equally between Gwen and Max, even though the total amount is an odd number.
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Rewrite the expression using the properties of exponents.
Using the properties of exponents we can write the given expression as [tex]x^\frac{3}{2} y^\frac{5}{2}z^\frac{-2}{3}[/tex]. Therefore, option E is the correct answer.
The given expression is [tex]\frac{\sqrt{x^3} \sqrt{y^5} }{\sqrt[3]{z^2} }[/tex].
We can square root as 1/2.
So the expression becomes [tex]\frac{x^\frac{3}{2} y^\frac{5}{2} }{z^\frac{2}{3} }[/tex]
= [tex]x^\frac{3}{2} y^\frac{5}{2}z^\frac{-2}{3}[/tex]
Therefore, option E is the correct answer.
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In a study of housing demand, the county assessor develops the following regression model to estimate the market value (i.e., selling price) of residential property within her jurisdiction. The assessor suspects that important variables affecting selling price (Y, measured in thousands of dollars) are the size of a house (X1, measured in hundreds of square feet), the total number of rooms (X2), age (X3), and whether or not the house has an attached garage (X4, No = 0, Yes 1). = Y = a + B1X1 + B2X2 + B3X3 + B4X4 + € = = = = -8.962, Now suppose that the estimate of the model produces following results: a = 182.550, bi = 4.067, b2 3.422, b3 0.019, b4 : 0.571, Sb2 = 3.225, Sb3 = 0.432, S14 = 7.302, R2 = 0.946, F-statistic = 43.61, and se = 7.923. Note that the sample consists of 15 randomly Sbl = = selected observations. According to the estimated model, holding all else constant, an additional year in age means the market value by approximately Which of the independent variables (if any) appears to be statistically significant (at the 0.05 level) in explaining the market value of residential property? Check all that apply. Size of the house (X1) Total number of rooms (X2) O Age (X3) Having an attached garage (X4) What proportion of the total variation in sales is explained by the regression equation? O 0.432 O 0.436 O 0.946 reject the null hypothesis that neither of the independent variables explains a significant (at The given F-value shows that the assessor the 0.05 level) proportion of the variation in income. Which of the following is an approximate 95 percent prediction interval for the selling price of a 15-year-old house having 16.00 hundred sq. ft., 3 rooms, and an attached garage? O (212.833, 244.525) O (233.365, 265.057) O (220.756, 236.602)
Previous question
According to the estimated model, holding all else constant, an additional year in age means the market value decreases by approximately $190 (0.019 x 1000).
The size of the house (X1) and the total number of rooms (X2) appear to be statistically significant (at the 0.05 level) in explaining the market value of residential property.
The proportion of the total variation in sales explained by the regression equation is 0.946.
The given F-value shows that the assessor can reject the null hypothesis that neither of the independent variables explains a significant (at the 0.05 level) proportion of the variation in income.
An approximate 95 percent prediction interval for the selling price of a 15-year-old house having 16.00 hundred sq. ft., 3 rooms, and an attached garage is (212.833, 244.525).
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In the expression 30+40+70, Jillian added 30 and 40 and then 70, while Samuel added 30 and 70 and then 40. Who is correct? Explain your reasoning
As per the mathematical operation both Jillian and Samuel are correct.
Given expression = 30+40+70,
The methodology by Jillian = added 30 and 40 and then 70
The methodology by Samuel = added 30 and 70 and then 40.
Determining the result of the given equation:
30 + 40 + 70
= 140
Reviewing the operations of both Jillian and Samuel
Jillian
He added 30 and 40 and then 70:
30 + 40 = 70
70 + 70 = 140
Samuel
He added 30 and 70 and then 40
30 + 70 = 100
100 + 40 = 140
The result of both mathematical operations is 140. Thus, it can be stated that both are correct.
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Farmer Fred was a quizzical man who loved to talk in circles. One day when a group of beginning farmers came to his farm, they asked him real nicely "farmer fred, how many chickens and rabbits do you have on your farm"? Being the quizzical guy that he was, he didn't dare give them a straight answer, but he reported "you know Im farmer Fren and what i`m all about, in this cage there are 24 heads and 36 feet use a system of equation to figure it out give 2 equations.
The number of chicken and the rabbits in the farm are 8 and 16 respectively.
Given that Fred has 24 heads and 80 feet in his farm we need to find the number of chicken and the rabbits in the farm,
Let the number of chicken and the rabbits in the farm be c and r respectively.
Establishing the system of equations,
c + r = 24
c = 24 - r.........(i)
2c + 4r = 80
c + 2r = 40
c = 40 - 2r..........(ii)
Equating the equation, since their LHS are equal,
24 - r = 40 - 2r
r = 16
Put r = 16 in eq(i)
c = 40 - 32
c = 8
Hence, the number of chicken and the rabbits in the farm are 8 and 16 respectively.
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Suppose we have converted the augmented matrix of a system of equations into reduced row-echelon form. How do we then identify the dependent and independent (free) variables?
We can easily identify the dependent and independent (free) variables by looking at the matrix after converting augmented matrix.
When we convert the augmented matrix of a system of equations into reduced row-echelon form, we can easily identify the dependent and independent (free) variables by looking at the matrix. The columns that contain leading 1's correspond to the variables that are dependent on the others. The columns that do not contain leading 1's correspond to the variables that are independent or free.
For example, if we have a matrix in reduced row-echelon form that looks like this:
1 0 2 | 3
0 1 -1 | 2
0 0 0 | 0
We can see that the first and second columns have leading 1's, so the variables corresponding to those columns (in this case, x1 and x2) are dependent on each other. The third column does not have a leading 1, so the variable corresponding to that column (in this case, x3) is independent or free.
In general, we can say that the number of independent variables is equal to the number of columns without leading 1's, and the number of dependent variables is equal to the number of columns with leading 1's. By identifying the dependent and independent variables, we can then use this information to solve the system of equations using substitution or elimination.
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Which equations represent circles that have a diameter of 12 units and a center that lies on the y-axis? Select two options. x2 + (y – 3)2 = 36 x2 + (y – 5)2 = 6 (x – 4)² + y² = 36 (x + 6)² + y² = 144 x2 + (y + 8)2 = 36
The equation of circles that have a diameter of 12 units and a center that lies on the y-axis is (x-0)² + (y-3)² = 6²
The standard form for finding the equation of a circle is expressed as;
(x-a)² + (y-b)²=r²
where;
(a, b) is the center of the circle
r is the radius of the circle
Given the following diameter = 12 units
radius = 12/2 = 6 units
If the centre lies on the y-axis, the center will be at (0, 3)
Substituting into the formula, we will have (x-0)² + (y-3)² = 6² equivalent to x²+ (y-3)² = 36
Hence, the equation of circles that have a diameter of 12 units and a center that lies on the y-axis is (x-0)² + (y-3)² = 6²
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I need help asap what’s the answer
Option B. Theresa's work is correct its shows that Matrix multiplication is not commutative.
Is matrix multiplication commutative?
Clearly, it is true that AB and BA are distinct matrices, so matrix multiplication is not generally commutable.
Nevertheless, there are special conditions where the product of matrices could be commutable, such as when one of them is an id matrix or diagonal matrix with equivalent diagonal entries.
Hence we can say that Theresa's work is correct its shows that Matrix multiplication is not commutative. since the results are different
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HELP ASAP 20 POINTS 7th grade math
a) The probability of picking a blue tile = 1/5
b) The probability of picking a yellow tile = 1/2
c) The probability of picking a red tile = 3/10
Given data ,
Let the total number of tiles in the bag = 10 tiles
The number of blue tiles = 2
The number of red tiles = 3
The number of yellow tiles = 5
So , probability of picking a blue tile = 2/10 = 1/5
Now , probability of picking a yellow tile = 5 / 10 = 1/2
And , probability of picking a red tile = 3/10
Hence , the probability is solved
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Many fingerprint analysts claim that they report an average of 16 matching characteristics between crime scene fingerprints and the suspect's fingerprints. In her last 30 cases, a defense attorney calculated that fingerprint analysts only reported an average of M = 9.8 matching characteristics, with a standard deviation of s = 1.2. What statistical test would this attorney use to determine whether the average number of matching characteristics she observed differs significantly from that reported by fingerprint analysts?
a. Independent-samples t test
b. Z test
c. Single-sample t test
d. Dependent-samples t test
Independent-samples t test. This test is used to compare the means of two independent groups. The correct answer is a.
In this case, the average number of matching characteristics reported by the fingerprint analysts and the average number of matching characteristics calculated by the defense attorney. The standard deviation of s = 1.2 would be used in the calculation of the t-statistic.
To determine whether the average number of matching characteristics the defense attorney observed (M = 9.8) differs significantly from that reported by fingerprint analysts (16 matching characteristics), she should use a "Single-sample t test".
Here's why:
- The defense attorney is comparing a sample (her 30 cases) to a known population mean (16 matching characteristics).
- The standard deviation of the sample is provided (s = 1.2), but the population standard deviation is not known.
- An Independent-samples t test would be used if comparing two independent groups, which is not the case here.
So, the correct answer is c. Single-sample t test.
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A right rectangular prism is sliced in a way such that the plane passes through three intersecting edges, cutting off a corner of the prism, what is the resulting cross section?
When a right rectangular prism is sliced in a way that the plane passes through three intersecting edges, cutting off a corner of the prism, the resulting cross section is a triangle.
Imagine a rectangular prism with the vertices A, B, C, D and H in order to understand this. Assume that the edges AE, and AG are crossed by the plane as it cuts off the corner indicated by vertex A. The triangle formed by the intersection of the plane and the prism will therefore have vertices at positions EFG.
The resulting cross section is a triangle EFG, which is a part of the original rectangular prism. The shape and size of the triangle will depend on the dimensions and orientation of the prism, as well as the specific location where the plane intersects the edges.
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*(Cauchy- Euler Equation): Find the general solution of the equation for the differential equation x^2y" + 4xy' -4y = 0 is (Select the correct answer) a. y = c_1x^-1 + c_2x^-1 ln x b. y = c_1x^-1 + c_2x^-2 c. y = c_1x^2 + c_2x d. y = c_1x + c_2x^-4 e. y = c_1x + c_2x^2
Given the following 2nd-order differential equation (Cauchy-Euler equation), [tex]x^2y''+4xy'-4y=0[/tex], find the general solution.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A differential equation in the form [tex]ax^2y''+bxy'+cy=0[/tex] , which is a Cauchy-Euler equation, can be solved in the following manner.
Where the characteristic equation => [tex]am^2+(b-a)m+c=0[/tex]
After solving for "m" the possible solutions are...
[tex]\bold{Real, \ Distinct \ Roots} \Rightarrow y=c_1x^{m_1}+c_2x^{m_2}\\\bold{ Repea ted \ Roots} \Rightarrow y=c_1x^{m}+c_2x^{m}ln(x)\\\bold{Complex \ Roots} \Rightarrow y=c_1x^{\alpha}cos(ln(x^\beta ))+c_2x^{\alpha}sin(ln(x^\beta )); \ m= \alpha \pm \beta i[/tex]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[tex]\Longrightarrow x^2y''+4xy'-4y=0 \ where \ a=1, \ b=4, \ and \ c=-4[/tex]
The characteristic equation:
[tex]\Longrightarrow (1)m^2+(4-1)m+(-4)=0 \Longrightarrow \boxed{m^2+3m-4=0}[/tex]
Solve for "m."
[tex]\Longrightarrow m^2+3m-4=0 \Longrightarrow (m-1)(m+4)=0 \Longrightarrow \boxed{m=1,-4}[/tex]
Note that m's are distinct and real.
[tex]Thus, \ \boxed{y=c_1x+c_2x^{-4}} \therefore Sol.[/tex]
Where the arbitrary constants c_1 and c_2 can be solved for given an initial condition.
The correct answer is option (e) [tex]$y = c_1 x + c_2 x^2$[/tex].
We can use the Cauchy-Euler equation to find the general solution of the given differential equation. The Cauchy-Euler equation is given by:
[tex]$a_n x^n y^{(n)}+a_{n-1} x^{n-1} y^{(n-1)}+\cdots+a_1 x y^{\prime}+a_0 y=0$[/tex]
where [tex]$a_n, a_{n-1}, \ldots, a_1, a_0$[/tex] are constants and [tex]$y^{(n)}, y^{(n-1)}, \ldots, y'$[/tex] denote the [tex]$n$[/tex]th, [tex]$(n-1)$[/tex]th, [tex]$\ldots$[/tex], first derivatives of y with respect to x.
In the given differential equation, we have [tex]$a_2 = 1$[/tex], [tex]$a_1 = 4$[/tex], and [tex]$a_0 = -4$[/tex]. Therefore, the Cauchy-Euler equation becomes:
[tex]$x^2 y^{\prime \prime}+4 x y^{\prime}-4 y=0$[/tex]
We assume a solution of the form [tex]$y=x^r$[/tex]. Substituting this into the differential equation, we get:
[tex]$$x^2 r(r-1) x^{r-2}+4 x r^1 x^{r-1}-4 x^r=0$$[/tex]
Simplifying, we get:
[tex]$$\begin{aligned}& r(r-1)+4 r-4=0 \\& r^2+3 r-4=0 \\& (r+4)(r-1)=0\end{aligned}$$[/tex]
So, we have two roots[tex]$r_1 = -4$[/tex] and [tex]$r_2 = 1$[/tex]. Therefore, the general solution of the differential equation is:
[tex]$y=c_1 x^{-4}+c_2 x^1$[/tex]
Thus, the correct answer is option (e) [tex]$y = c_1 x + c_2 x^2$[/tex].
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TRUE OR FALSE if a line passes through the points (20,5) and (10,10), then the slope of the line is -2.
False. The slope of the line is actually -0.5.
Let's determine the slope of the line that passes through the points (20, 5) and (10, 10). To find the slope, we can use the formula:
Slope (m) = (y2 - y1) / (x2 - x1)
Plugging in the coordinates of the two points, we get:
m = (10 - 5) / (10 - 20) = (5) / (-10) = -1/2
The slope of the line is -1/2. Therefore, the statement "If a line passes through the points (20, 5) and (10, 10), then the slope of the line is -2" is FALSE.
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