1. 18
2. 26
3. 25
1. 3×5+3= 18
2. 4×5+2×3= 26
3. 5×5+3×3-9= 25
When a number and letter are next to each other mean multiply (Example)
5a = 5×a
A circle has a circumference of 20 ft. What is the
radius of the circle?
Step-by-step explanation:
The formula to calculate the circumference of a circle is:
Circumference = 2πr
where "r" is the radius of the circle and "π" is the mathematical constant pi, approximately equal to 3.14159.
Given that the circumference of the circle is 20 ft, we can plug in the values into the formula and solve for "r":
20 = 2πr
Divide both sides of the equation by 2π:
r = 20 / (2π)
Simplify the expression by using the approximation of π as 3.14159:
r = 20 / (2 x 3.14159)
r = 3.1831 ft (rounded to four decimal places)
Therefore, the radius of the circle is approximately 3.1831 feet.
The battery life on Marco's cell phone decreases
according to the scatterplot below. What is the longest
amount of time Marco could use his phone without
charging it?
Answer:
The longest amount of time that Marco could use his phone without changing the battery is given as follows:
3.3 hours.
How to define a linear function?The slope-intercept representation of a linear function is given by the equation presented as follows:
y = mx + b
The coefficients of the function and their meaning are described as follows:
m is the slope of the function, representing the change in the output variable y when the input variable x is increased by one.b is the y-intercept of the function, which is the initial value of the function, i.e., the numeric value of the function when the input variable x assumes a value of 0. On a graph, it is the value of y when the graph of the function crosses tbe y-axis.In 5 hours, the battery fell from 50% to 20%, hence the slope is given as follows:
-30/5 = -6% an hour.
The battery is currently at 20%, hence the time that Marco could use the phone without charging it is given as follows:
20/6 = 3.3 hours.
More can be learned about linear functions at https://brainly.com/question/24808124
#SPJ1
The following question has 2 parts. First, answer part A. Then, answer part B.
Use the image below to answer the questions.
Help fast please!
The fraction that represents the addition model of part A is 5/10, and for part B is 3/100.
What is a fraction?Parts of a whole are defined as fractions. The top and bottom numbers of a fraction are explained in the fundamentals of fractions. The number at the top represents the number of selected pieces of a whole, while the number at the bottom reflects the entire number of components.
According to the given information:
Based on the information provided, a grid A with 10 equal columns is depicted, with 5 shaded in. This will be expressed as a score of 5/10.
A second grid B, same in size, with 100 squares shaded in is also presented. This will be expressed as 3/100.
In this situation, Emma's response is erroneous because he added 5 to 3 and received an 8/100.
To know more about Fraction visit:
brainly.com/question/78672
#SPJ1
Find the x and y-intercept and Find the vertical and horizontal asymptotes
r(x)= x^2 + 3x - 10/ (x+1)(x-3)(x+5)
The y-intercept of the equation is -10.
What is Equation?Equation is a mathematical statement that expresses the equality of two expressions, which are separated by an equals sign. It can be used to represent a variety of different relationships, such as equality, inequality, addition, subtraction, multiplication, and division. Equations can be used to solve a variety of problems in science, engineering, and mathematics.
The x-intercepts of the graph of the equation r(x) are those values of x for which the equation is equal to 0. To find the x-intercepts, we must solve the equation:
x²+ 3x - 10 = 0
Factoring the equation, we get:
(x+5)(x-3)(x+1) = 0
The solutions of this equation are x = -5, -1, and 3. Therefore, the x-intercepts of the equation are -5, -1, and 3.
To find the y-intercept, we must plug 0 in for x in the equation:
y = x² + 3x - 10/ (x+1)(x-3)(x+5)
When x = 0, we get:
y = -10/ (0+1)(0-3)(0+5)
Therefore, the y-intercept of the equation is -10.
The vertical asymptote of the equation is the line x = -1, since it is the only value of x that makes the denominator equal to 0.
The horizontal asymptote of the equation is the line y = 0, since the degree of the numerator is less than the degree of the denominator.
To know more about Equation click-
https://brainly.com/question/2972832
#SPJ1
The y-intercept of the equation is -10.the x-intercepts of the equation are -5, -1, and 3.
What is Equation?Equation is a mathematical statement that expresses the equality of two expressions, which are separated by an equals sign. It can be used to represent a variety of different relationships, such as equality, inequality, addition, subtraction, multiplication, and division. Equations can be used to solve a variety of problems in science, engineering, and mathematics.
The x-intercepts of the graph of the equation r(x) are those values of x for which the equation is equal to 0. To find the x-intercepts, we must solve the equation:
x²+ 3x - 10 = 0
Factoring the equation, we get:
(x+5)(x-3)(x+1) = 0
The solutions of this equation are x = -5, -1, and 3. Therefore, the x-intercepts of the equation are -5, -1, and 3.
To find the y-intercept, we must plug 0 in for x in the equation:
y = x² + 3x - 10/ (x+1)(x-3)(x+5)
When x = 0, we get:
y = -10/ (0+1)(0-3)(0+5)
Therefore, the y-intercept of the equation is -10.
The vertical asymptote of the equation is the line x = -1, since it is the only value of x that makes the denominator equal to 0.
The horizontal asymptote of the equation is the line y = 0, since the degree of the numerator is less than the degree of the denominator.
To know more about Equation click-
brainly.com/question/2972832
#SPJ1
A rectangular bathroom tile is 2 and 1/3 times as wide as it is tall . If the tile is 5 cm tall,how wide is is
A 3 and 2/3 cm
B 7 and 1/3 cm
C 10 and 1/3 cm
Help me find the answer pls
The width of the rectangular tile is obtained to be 11 and 2/3 cm.
What is a rectangle?
A quadrilateral with parallel sides that are equivalent to one another and four equal vertices is known as a rectangle. It is also known as an equiangular rectangle for this reason.
The bathroom tile is in a rectangular shape.
Let's call the width of the tile "w".
From the problem, we know that -
w = (2 and 1/3) × h
We also know that the height of the tile is 5 cm.
Substituting this value in the equation above, we get -
w = (2 and 1/3) × 5 cm
w = 7/3 × 5 cm
w = 35/3 cm
w = 11 and 2/3 cm
Therefore, the width of the tile is 11 and 2/3 cm.
To learn more about rectangle from the given link
https://brainly.com/question/2607596
#SPJ1
A family drives 509 miles to their vacation destination. It takes them 8.6 hours to get there. What was their average velocity in miles per hour?
Answer:
Their average velocity during their trip was 59.19 miles per hour.
Step-by-step explanation:
To calculate the average velocity of the family's trip, we divide the total distance traveled by the time it took to travel that distance:
Average velocity = total distance ÷ time
In this case, the total distance traveled is 509 miles and the time it took to travel that distance is 8.6 hours. So, we can plug these values into the formula:
Average velocity = 509 miles ÷ 8.6 hours
Simplifying this equation, we get:
Average velocity = 59.19 miles per hour (rounded to two decimal places)
Therefore, the family's average velocity during their trip was 59.19 miles per hour.
A secretary listed the frequency of calls by a patient to a doctor's office during a year, and the data is displayed in the histogram.
A histogram titled Calls To A Doctor. The x-axis is labeled Number of Calls and has intervals listed for 1 to 25, 26 to 50, 51 to 75, and 76 to 100. The y-axis is labeled Frequency and begins at 0, with tick marks every five units up to 60. There is a shaded bar for 1 to 25 that stops at 58, for 26 to 50 that stops at 34, for 51 to 75 that stops at 17, and for 76 to 100 that stops at 12.
Which of the following best describes the spread and distribution of the data?
The data is skewed to the higher values because many patients call the doctor several times a day when they are sick.
The data is skewed to the lower values because many patients only call the doctor once or twice a year when they are sick.
The data is bimodal with a narrow spread because many patients call the doctor several times a day when they are sick.
The data is normal and has a wide spread because many patients only call the doctor once or twice a year when they are sick.
The data is skewed to the lower values because many patients only call the doctor once or twice a year when they are sick.
Define the terms spread and distribution?Spread and distribution of data are statistical concepts that describe how a set of data is spread out or distributed.
Distribution refers to how the data is distributed or arranged. It is typically described in terms of its shape, center, and spread. Common types of distributions include normal distribution, skewed distribution, and uniform distribution. The distribution can be visualized using various graphs such as histograms, box plots, and scatter plots.
Based on the histogram description, the best answer is:
The data is skewed to the lower values because many patients only call the doctor once or twice a year when they are sick.
The histogram shows that the frequency of calls is highest for the 1 to 25 interval, with decreasing frequency for each subsequent interval. The fact that the shaded bars stop at relatively low frequencies for each interval suggests that there are very few patients who call the doctor frequently, skewing the data towards the lower values. The histogram also shows a wide range of values, or spread, with the highest interval going up to 100, but the low frequencies in each interval suggest that the data is not normally distributed.
To know more about set of data, visit:
https://brainly.com/question/27358262
#SPJ1
Answer:its b
Step-by-step explanation:
There are 88 seats in a theater. The seating in the theater is split into 4 identical sections. Each section has 14 red seats and some blue seats.
write an exspression
Answer:
x = 8
Step-by-step explanation:
Let x be the number of blue seats in each section.
Then the total number of seats in each section is:
14 + x
Since there are 4 identical sections, the total number of seats in the theater is:
4(14 + x) = 56 + 4x
We know that the theater has 88 seats, so:
56 + 4x = 88
Simplifying the equation:
4x = 32
x = 8
Therefore, each section has 14 red seats and 8 blue seats.
The expression for the number of blue seats in each section is x = 8.
Answer:
4(14 + b) = 88---------------------------------
Let b be the number of blue seats.
There are 4 sections, each has 14 red and b blue seats, with the total number of seats being 88.
The equation to reflect this would be:
4(14 + b) = 88Its solution would be:
14 + b = 88/414 + b = 22b = 22 - 14b = 8Use the graph to answer the question. Graph of a polygon ABCD with vertices at 4 comma 4, 12 comma 4, 12 comma 8, 4 comma 8 and a second polygon A prime B prime C prime D prime with vertices at 1 comma 1, 3 comma 1, 3 comma 2, 1 comma 2. Determine the scale factor used to create the image. one half 2 one fourth 2.5
The image was created by translating 5 units down, 5 units up, 1 unit down, and 1 unit up. ABCD and A'B'C'D' are two polygons on the graph.
The first polygon, ABCD, has vertices at (4,4), (12,4), (8,4), and (1,1),). (3,1). The vertices of the second polygon, A'B'C'D', are located at (1,1), (3,1), (3,2), and (1,2). (0.5,.25).
We can establish the translation that was applied to produce the image by comparing the coordinates of the two polygons. The two polygons' initial coordinate is (1,-1) and their second coordinate is (1,-1). (1,-6). The translation is 5 units lower because of the 5-unit difference in the y-coordinate. The second coordinate for the two polygons is (3,-5) and (3,-10). The translation is 5 units lower because of the 5-unit difference in the y-coordinate.
The third coordinate of the second polygon is (7,-5) and (7,-10). The translation is 5 units lower because of the 5-unit difference in the y-coordinate. The fourth coordinate of both polygons is (5,-1). (5,-6). The translation is 5 units lower because of the 5-unit difference in the y-coordinate.
The translation of the image was five units down, five units up, one unit down, and one unit up.
For more questions related to coordinates,
brainly.com/question/13662452
#SPJ1
Answer: C = 1/4
Step-by-step explanation: I got it right on my test
Hope this helps :)
Diane bought 10 pounds of sugar for $5.
How many pounds of sugar did she get per dollar?
Answer:
Diane got 2 pounds of sugar per dollar.
To see why, divide the number of pounds of sugar by the cost in dollars:
10 pounds / $5 = 2 pounds/dollar
From a 2-pound bag of flour, Marcella took 1/4 pound to bake bread. Which expression tells the weight of the flour left in the bag?
2-0.25
2-1.4
2-0.14
2-0.025
2.5-2
The expressiοn that tells the weight οf the flοur left in the bag is 2 - 0.25
What is an expressiοn?A finite cοmbinatiοn οf symbοls that are well-fοrmed in accοrdance with cοntext-dependent principles is referred tο as an expression or mathematical expression. Unknown variables, integers, and arithmetic operators are the components of an algebraic expression.
Here, we have
Given: From a 2-pound bag of flour, Marcella took 1/4 pound to bake bread.
We have to find the expression that tells the weight of the flour left in the bag.
Marcella took 1/4 pound to bake bread = 0.25
If Marcella took 0.25 pounds from a 2-pound flour bag then she left with 1.75 pounds.
Hence, the expression that tells the weight of the flour left in the bag is 2 - 0.25
To learn more about the expression from given link
https://brainly.com/question/1859113
#SPJ1
Divide 12a7b2 by 4a2b. 3a5b2 3a5b3 3a5b 3a9b3
Answer: 3a^5b
Step-by-step explanation:
I just did an exam with the question, and I got it correct.
Answer:
Like the other person said, C.
Step-by-step explanation:
also got it correct on the test
have a good day xx
For every pound of warming gas produced by agriculture, how many pounds of warming gas are produced by transportation
1.0
1.3
2.0
2.7
3.0
The correct option is (e) i.e. 3.0.
What are Units ?
In math, units refer to a standard of measurement used to quantify a quantity. Units are used to specify the magnitude of a physical quantity or measurement, such as length, mass, time, temperature, and so on.
The use of units is important in math and science because it allows us to communicate and compare measurements effectively. Without units, it would be difficult to understand or compare different measurements, as they would lack a common reference point.
According to the United States Environmental Protection Agency (EPA), transportation is the largest contributor to greenhouse gas emissions in the United States, responsible for approximately 29% of total greenhouse gas emissions in 2019. In comparison, agriculture accounted for approximately 10% of total greenhouse gas emissions in the same year.
Therefore, for every pound of warming gas produced by agriculture, approximately 2.9 pounds of warming gas are produced by transportation.
So, the closest option from the given choices is 3.0.
To learn more about Units, visit the link:
https://brainly.com/question/28464
#SPJ1
I need help question
Answer:
a= positive
b= When the graph goes up it means it has a positive trend.
c= the direction does not make sense because the measurements for the x and y coordinate do not correlate with each other.
Step-by-step explanation:
What would (-2, -2) be if it is rotated 90 degrees clockwise?
Answer:
(2, -2)
Step-by-step explanation:
To rotate a point in a coordinate plane 90 degrees clockwise, we need to switch the x and y coordinates and negate the new x coordinate.
So let's apply this formula to the point (-2, -2):
- The new x-coordinate will be the original y-coordinate: -2
- The new y-coordinate will be the negation of the original x-coordinate: -(-2) = 2
Therefore, after rotating (-2, -2) 90 degrees clockwise, we get the point (2, -2).
We can visualize this by plotting both points on a coordinate plane and drawing a line of reflection that represents the rotation:
```
Before rotation:
|
|
|
-------+-------
|
(-2,-2)
After rotation:
|
|
(2,-2)|
-------+-------
|
```
As shown in the diagram above, after rotating (-2,-2) 90 degrees clockwise, it lands at (2,-2).
Which of the following is the same as 8(9+2) ?
8*9+8
8*9+2
8*9+8*2
9(8+2)
2(8+9)
According the question 89+82 expressiοn is the same as 8(9+2).
What is expressiοn?An expressiοn in math is a sentence with a minimum οf twο numbers οr variables and at least οne math οperatiοn. This math οperatiοn can be additiοn, subtractiοn, multiplicatiοn, οr divisiοn.
Algebraic expressiοns are classified οn the basis οf the number οf terms in the expressiοn. The variοus types οf algebraic expressiοns are:
Mοnοmial expressiοnsBinοmial expressiοnsTrinοmial expressiοnsPοlynοmial expressiοnsThe difference between expressiοns and equatiοns is that an expressiοn signifies a cοmbinatiοn οf numbers, variables, and οperatiοn symbοls whereas an equatiοn will always use an equal (=) οperatοr between twο math expressiοns.
The expressiοn 8(9+2) can be simplified using the distributive prοperty οf multiplicatiοn οver additiοn:
8(9+2) = 89 + 82
Therefοre, the answer is: 89+82
To know more about expression visit,
https://brainly.com/question/1859113
#SPJ1
Answer this question please
The transformation that it could have been is either a rotation or a reflection, over one of the vertices of the figure.
What are transformations on a figure?The possible transformations on a figure are listed as follows:
Translation: Translation left/right or down/up, moving all vertices.Reflections: Over one of the axes or over a line.Rotations: Over a degree measure.Dilation: Coordinates of the vertices of the original figure are multiplied by the scale factor, hence all vertices are changed.Either a reflection or a rotation can happen over one of the vertices of the graph, which will keep it's coordinates constant, and thus the transformation in the context of this problem is either one of these two.
More can be learned about transformations at https://brainly.com/question/29209050
#SPJ1
I will mark you brainiest!
Given square ERTN, what is the length of NT ?
A) 50
B) 15
C) 25
Answer:
C) 25
Step-by-step explanation:
A square has all four sides equal.
5x = 10x - 25
-5x = -25
x = 5
Now put 5 in for x, and you get the answer is 25
A high school drama club is selling tickets for the spring production. One day the president sold 12 adult tickets and 5 child tickets for a total of $169. The next day the president sold 22 adult tickets and 10 child tickets for a total of $314. Let a represent the cost of one adult ticket. Let c represent the cost of one child ticket. Which system of equations will solve for a and c?
12 a + 5 c = 314. 22 a + 10 c = 169.
12 a + 5 c = 169. 22 a + 10 c = 314.
5 a + 12 c = 314. 22 a + 10 c = 169.
22 a + 5 c = 169. 12 a + 10 c = 314.
Mark this and return
The cοst οf οne adult ticket is $12 and the cοst οf οne child ticket is $5.
What is system οf equatiοn?A system οf linear equatiοns can be sοlved graphically, by substitutiοn, by eliminatiοn, and by the use οf matrices.
The cοrrect system οf equatiοns tο sοlve fοr a and c is:
12a + 5c = 169 (equatiοn 1)
22a + 10c = 314 (equatiοn 2)
These equatiοns represent the tοtal ticket sales and revenue οn twο different days.
Tο sοlve this system, we can use the methοd οf eliminatiοn.
-24a - 10c = -338
22a + 10c = 314
-2a = -24
Dividing bοth sides by -2:
a = 12
Nοw we can substitute a = 12 intο equatiοn 1 tο sοlve fοr c:
12(12) + 5c = 169
144 + 5c = 169
5c = 25
c = 5
Therefοre, the cοst οf οne adult ticket is $12 and the cοst οf οne child ticket is $5.
To know more about system of equation visit,
https://brainly.com/question/13729904
#SPJ1
Which statement can you conclude from the given information?
Given: Points X, Y, and Z are equidistant from points P and Q.
O Point Y is equidistant from points X and Z.
OPX = PY
O PQ=XZ
O Points X, Y, and Z are collinear.
Please help
Therefore , the solution of the given problem of equation comes out to be Y is equally distant from X and Z.
What is equation?Variable words are commonly used in complex algorithms to show consistency between two contradictory claims. Academic expressions called equations are used to show the equality of various academic numbers. In this case, normalization produces a + 7 as opposed to a different algorithm that splits 12 into two parts and can evaluate data received from x + 7.
Here,
We can infer from the knowledge that
=> PQ = XZ
because X, Y, and Z are equally spaced apart from P and Q.
Based on this knowledge alone, we cannot infer that X, Y, and Z are collinear or that Y is equally distant from X and Z.
Therefore , the solution of the given problem of equation comes out to be Y is equally distant from X and Z.
To know more about equation visit:
https://brainly.com/question/649785
#SPJ1
The city of Pineville held a talent contest. Judges rated the contestants on a scale from 1 to
10. A singer was rated 7.4, while a juggler was rated 7.0. The median rating was 7.9, and the
interquartile range was 0.8. The winner was a magician who was rated 9.7.
Which number best describes a typical rating in the contest?
0.8
7.4
7.9
9.7
The answer of the given question based on the number best describes a typical rating in the contest the answer is 7.9.
What is Median?The median is measure of central tendency in statistics that represents middle value of dataset. It is value that separates highest 50% of values from the lowest 50%.
To find median of a dataset, you first need to sort data in ascending or descending order.
The median rating is 7.9, which means that half of the contestants scored above 7.9 and half scored below. This suggests that 7.9 is a representative value of the ratings overall.
Additionally, the interquartile range (IQR) is a measure of the spread of the middle 50% of the ratings, and it is given as 0.8. This means that the ratings between the 25th and 75th percentile fall within a range of 0.8, which suggests that the ratings are fairly tightly clustered around the median of 7.9.
Therefore, the number that best describes a typical rating in the contest is 7.9.
To know more about Dataset visit:
https://brainly.com/question/30154121
#SPJ1
What is the perimeter of the rectangle in the coordinate plane?
_______________________________
(reporting spams/wrong answers)
(giving brainliest to the correct answer)
_______________________________
Answer:
40 units
Step-by-step explanation:
12+12+8+8=40 units
can you please solve this its urgent
Step-by-step explanation:
Think we can get a better picture, maybe more light? I wouldn't mind helping if you can
c) What is the height of the container if it’s a cylinder with a radius of 5cm?
d) If the contrainer from Part C is full of Hot Chocolate powder and Hot Chocolate powder costs $0.07 per cubic centimeter, what is the cost of the hot chocolate powder?
The height of this container if it is a cylinder with a radius of 3 cm is equal to 5 cm.
The cost of coffee is equal to $2.83.
The height of this container if it is a cylinder with a radius of 5 cm is equal to 1.8 cm.
The cost of hot chocolate powder is equal to $9.90.
How to calculate the volume of a cylinder?In Mathematics, the volume of a cylinder can be determined by using the following formula:
Volume of a cylinder, V = πr²h
Where:
V represents the volume of a cylinder.h represents the height of a cylinder.r represents the radius of a cylinder.By choosing a volume of 45π cm³, the height of this container can be calculated as follows:
h = V/πr²
Height, h = 45π/π3²
Height, h = 5 cm.
For the cost of coffee, we have:
Cost of coffee = 45π cm³ × $0.02
Cost of coffee = $2.83.
When radius, r = 5 cm, the height of this container is:
h = V/πr²
Height, h = 45π/π5²
Height, h = 1.8 cm.
For the cost of the hot chocolate powder, we have:
Cost of the hot chocolate powder = 45π × $0.07
Cost of the hot chocolate powder = $9.90.
Read more on cylinder here: brainly.com/question/14060443
#SPJ1
Online Content: Site 1
When calculating simple interest, what must you do if you want to invest for months instead of years?
Which name is a correct way to name the tiling?
Is -3 a real irrational rational integer or a whole number 
-3 is not an irrational number, but it is a real number, a rational number, an integer, and a whole number.
What are whole numbers and rational numbers?
All numbers, including integers, fractions, decimals, and irrational numbers, that can be represented on a number line are considered real numbers.
In summary, real numbers include rational and irrational numbers, and integers include whole number and their negatives.
When two integers are compared, a number can be stated as a ratio, such as -3/1, 1/2, or 5/7. Writing rational numbers as recurring or ending decimals is an option.
Numbers that can't be stated as a ratio of two integers, like 2 or, are referred to as irrational numbers. It is impossible to write irrational numbers using repeating or ending decimals.
Positive and negative whole numbers, as well as zero, are considered integers. Since whole numbers are non-negative integers, they contain zero and only positive values.
Learn more about Decimals here:
brainly.com/question/29765582
#SPJ9
Given that Z=√3 + √3 i, write Z in Euler form
Answer:
To write Z in Euler form, we first need to find its magnitude and argument.
The magnitude of Z is given by:
|Z| = √(Re(Z)^2 + Im(Z)^2)
where Re(Z) is the real part of Z, and Im(Z) is the imaginary part of Z.
In this case:
Re(Z) = √3
Im(Z) = √3 i
So:
|Z| = √(√3^2 + (√3)^2) = √(3 + 3) = √6
The argument of Z is given by:
arg(Z) = tan^(-1)(Im(Z) / Re(Z))
In this case:
arg(Z) = tan^(-1)((√3) / √3) = tan^(-1)(1) = π/4
Therefore, Z in Euler form is:
Z = |Z| e^(i arg(Z)) = √6 e^(i π/To write Z in Euler form, we first need to find its magnitude and argument.
The magnitude of Z is given by:
|Z| = √(Re(Z)^2 + Im(Z)^2)
where Re(Z) is the real part of Z, and Im(Z) is the imaginary part of Z.
In this case:
Re(Z) = √3
Im(Z) = √3 i
So:
|Z| = √(√3^2 + (√3)^2) = √(3 + 3) = √6
The argument of Z is given by:
arg(Z) = tan^(-1)(Im(Z) / Re(Z))
In this case:
arg(Z) = tan^(-1)((√3) / √3) = tan^(-1)(1) = π/4
Therefore, Z in Euler form is:
Z = |Z| e^(i arg(Z)) = √6 e^(i π/4)
Factor completely using the GCMF.
10x5 + 8x4 - 4x²
Answer:
To factor completely using the GCMF (Greatest Common Monomial Factor), we first need to identify the common monomial factor of all three terms. In this case, the common monomial factor is 2x²:
10x^5 + 8x^4 - 4x² = 2x²(5x^3 + 4x^2 - 2)
Now, we can see that the expression can be factored further, but it cannot be factored using the GCMF. However, we can use other factoring techniques, such as grouping or factoring by grouping, to factor the remaining expression:
2x²(5x^3 + 4x^2 - 2) = 2x²(5x^3 + 10x^2 - 6x^2 - 2)
= 2x²(5x^2(x + 2) - 2(x + 2))
= 2x²(5x^2 - 2)(x + 2)
Therefore, the expression 10x^5 + 8x^4 - 4x² can be factored completely as 2x²(5x^2 - 2)(x + 2).
Step-by-step explanation:
Find the two linearly independent solutions of the LODE following for x>0, and determine at least the first four leading terms in the second solution y(2). 9x²y" - 6xy' + 2y = 0
Please, I need the solution
Using Cauchy-Euler equation, the solution of the second-order homogeneous differential equation is y2(x) = c3 * x^(1/3) + c4 * x^(4/3) - (1/2) * c4 * x
What is the solution to the equationThe given LODE is a second-order homogeneous differential equation with constant coefficients. The characteristic equation is given by:
9r^2 - 6r + 2 = 0
Using the quadratic formula, we find the roots to be:
r = (1/3) ± i/sqrt(3)
Thus, the general solution to the differential equation is:
y(x) = c1 * x^(1/3) * cos((1/3)*arccos(sqrt(3)*r)) + c2 * x^(1/3) * sin((1/3)*arccos(sqrt(3)*r))
To find the first solution, we can choose either the cosine or sine term. Let's choose the cosine term for simplicity:
y1(x) = x^(1/3) * cos((1/3)*arccos(sqrt(3)*r))
To find the second solution, we can use the method of reduction of order. Assume that the second solution has the form:
y2(x) = v(x) * y1(x)
Substituting this into the differential equation, we obtain:
9x^2(v''(x)*y1(x) + 2v'(x)*y1'(x) + v(x)*y1''(x)) - 6x(v'(x)*y1(x) + v(x)*y1'(x)) + 2v(x)*y1(x) = 0
Simplifying this equation by dividing by x^2*y1(x), we get:
9v''(x) + (18/r)x*v'(x) + ((2/r^2) - 1)v(x) = 0
where r = sqrt(3)
This is a Cauchy-Euler equation, which can be solved by assuming a solution of the form:
v(x) = x^m
Substituting this into the equation, we obtain:
9m(m-1) + (18/r)m + ((2/r^2) - 1) = 0
Simplifying this equation, we get:
9m^2 + 6m + 1 = 0
Using the quadratic formula, we find the roots to be:
m = (-1/3) or (-1/3)
Thus, the second solution is given by:
y2(x) = c3 * x^(1/3) + c4 * x^(1/3) * ln(x)
To determine the first four leading terms in the second solution, we can use the Taylor series expansion of the natural logarithm:
ln(x) = (x-1) - (1/2)(x-1)^2 + (1/3)(x-1)^3 - ...
Substituting this into the second solution, we obtain:
y2(x) = c3 * x^(1/3) + c4 * x^(1/3) * [(x-1) - (1/2)(x-1)^2 + (1/3)(x-1)^3 - ...]
Simplifying this expression, we get:
y2(x) = c3 * x^(1/3) + c4 * x^(1/3) * (x-1) - c4 * x^(1/6) * (1/2)(x-1)^2 + c4 * x^(1/3) * (1/3)(x-1)^3 - ...
Thus, the first four leading terms in the second solution are:
y2(x) = c3 * x^(1/3) + c4 * x^(4/3) - (1/2) * c4 * x
learn more on Cauchy-Euler equation here;
https://brainly.com/question/15518645
#SPJ1