Points a, b, and c are collinear. If bc = 8. 5 and ac = 13. 2, select all possible values of ab.

First we have to factorise the denominator [tex]\rm{x^3+1.}[/tex]

It can be factorised as,

[tex]\rm{x^3+1=(x+1)(x^2-x+1)}[/tex]

Then the fraction can be written as,

[tex]\longrightarrow\rm{\dfrac{x^2+1}{x^3+1}=\dfrac{x^2+1}{(x+1)(x^2-x+1)}\quad\dots(1)}[/tex]

This fraction can be resolved into partial fractions by writing it as the sum of a fraction having denominator [tex]\rm{x+1}[/tex] and a fraction having denominator [tex]\rm{x^2-x+1.}[/tex]

Since [tex]\rm{x+1}[/tex] is a first degree polynomial, the numerator of the fraction having this term as denominator should be a zero degree polynomial, or a constant term, assumed as [tex]\rm{A.}[/tex]

Since [tex]\rm{x^2-x+1}[/tex] is a second degree polynomial, the numerator of the fraction having this term as denominator should be a first degree polynomial, assumed as [tex]\rm{Bx+C.}[/tex]

So let,

[tex]\longrightarrow\rm{\dfrac{x^2+1}{(x+1)(x^2-x+1)}=\dfrac{A}{x+1}+\dfrac{Bx+C}{x^2-x+1}}[/tex]

[tex]\small\text{$\longrightarrow\rm{\dfrac{x^2+1}{(x+1)(x^2-x+1)}=\dfrac{A(x^2-x+1)+(Bx+C)(x+1)}{(x+1)(x^2-x+1)}}$}[/tex]

Equating numerators,

[tex]\longrightarrow\rm{x^2+1=A(x^2-x+1)+(Bx+C)(x+1)}[/tex]

[tex]\small\text{$\longrightarrow\rm{x^2+1}=\cal{(A+B)}\rm{x^2}+\cal{(-A+B+C)}\rm{x}+\cal{(A+C)}$}[/tex]

Equating corresponding coefficients,

[tex]\rm{A+B=1}[/tex]

[tex]\rm{-A+B+C=0}[/tex]

[tex]\rm{A+C=1}[/tex]

Solving these three equations we get,

[tex]\rm{A=\dfrac{2}{3}}[/tex]

[tex]\rm{B=\dfrac{1}{3}}[/tex]

[tex]\rm{C=\dfrac{1}{3}}[/tex]

Thus,

[tex]\small\text{$\longrightarrow\rm{\dfrac{x^2+1}{(x+1)(x^2-x+1)}=\dfrac{2}{3(x+1)}+\dfrac{x+1}{3(x^2-x+1)}}$}[/tex]

[tex]\small\text{$\longrightarrow\rm{\dfrac{x^2+1}{(x+1)(x^2-x+1)}=\dfrac{1}{3}\left[\dfrac{2}{x+1}+\dfrac{x+1}{x^2-x+1}\right]}$}[/tex]

So (1) becomes,

[tex]\longrightarrow\underline{\underline{\bf{\dfrac{x^2+1}{x^3+1}=\dfrac{1}{3}\left[\dfrac{2}{x+1}+\dfrac{x+1}{x^2-x+1}\right]}}}[/tex]

Hence the given fraction is resolved into partial fractions.

Mean =  (sum of all observations in given data set) ÷ (total number of observations)

In an odd-numbered set, the median will be the number in the very middle of the list. In an even-numbered set, the average of the two middle numbers will be mode.

The mode is the most frequent number.

Mean:

Using two straightforward procedures, we can determine the mean, or average, of a data set:

(1) Add up all of the values to find the sum.

(2) Divide the sum by the number of values in the data set

Mean = (sum of all observations in given data set) ÷ (total number of observations)

Median:

The middle score in the group is known as the median. To find the median, start by arranging all of the data points from smallest to largest. Means arrange the data either in ascending order or in the descending order. In an odd-numbered set, the median will be the number in the very middle of the list. In an even-numbered set, you will need to calculate the average of the two middle numbers.

Mode:

To quickly determine the mode, arrange the numbers in ascending order by least to greatest, then tally the frequency of each number. The mode is the most frequent number.

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(a) sample proportion is 0.75.

(b) The 99% confidence interval for the population proportion is approximately [0.685, 0.815].

(c) a 99% probability that the true population proportion of voters who would support Maria Wilson is within this range.

What is proportion?

A proportion in math is when two ratios are equivalent, that means they are equal to each other.

(a) To calculate the sample proportion, we need to divide the number of voters who would support Maria Wilson by the total sample size. In this case, the sample proportion is 300/400 = 0.75.

(b) To develop a 99% confidence interval for the population proportion, we can use the following formula:

confidence interval = sample proportion +/- (z-value * standard error)

The standard error of the sample proportion is a measure of the variability of the sample proportion, and is calculated as:

[tex]standard\ error = \sqrt{((sample\ proportion * (1 - sample\\ proportion)) / sample size)}\\\\standard\ error= \sqrt{((0.75 * 0.25) / 400)}\\\\standard\ error= 0.025[/tex]

At a 99% confidence level, the z-value is approximately 2.576.

Substituting these values into the formula for the confidence interval, we get:

confidence interval = 0.75 +/- (2.576 * 0.025)

= 0.75 +/- 0.065

= [0.685, 0.815]

So, The 99% confidence interval for the population proportion is approximately [0.685, 0.815].

(c) The confidence interval tells us that we are 99% confident that the population proportion of voters who would support Maria Wilson in the November election is between 0.685 and 0.815. This means that there is a 99% probability that the true population proportion of voters who would support Maria Wilson is within this range.

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9/10 is the quotient when 3/4 is divided by 5/6.

The two parts that make up a fraction are the numerator, which is located at the top, and the denominator, which is located at the bottom.

When two fractions are divided, the first fraction is multiplied by the second fraction's reciprocal. Finding the reciprocal of the second fraction and this can be obtained by reversing its numerator and denominator—is and then dividing the  fractions. Multiply the two numerators after that. Multiply the two denominators after that, After that we need to simplify the fraction to its lowest form.

Mathematically quotient ,

= (3/4) ÷ (5/6)

It is possible to define an arithmetic operation as one in which we add, subtract, multiply, and divide numbers. It has the fundamental operators +, -,÷,× and.

= (3/4)*(6/5)

= 18/20

= 9/10

so the quotient of 3/4 and 5/6 in lowest form is 9/10.

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The mean and standard deviation of the distribution of R-S is

The mean is calculated by simply substracting the values

= mean R - means S

= 10 - 7

= 3

standard deviation, SD

standard deviation is solved by the use of radicals and this affects the solution hence it will not be simple subtraction, rather we solve back to remove the radical and hence we get the variance

= (SD, R)² + (SD, S)²

= 4² + 3²

= 16 + 9

= 25

the variance is 25

SD = √variance = √25 = 5

complete question

The distribution of random variable R has mean 10 and standard deviation 4. The distribution of random variable S

has mean 7 and standard deviation 3. If Rand S are independent, what are the mean and standard deviation of the

distribution of R-S?

A Mean 3 and standard deviation 1

B Mean 3 and standard deviation 5

C Mean 3 and standard deviation 7

D Mean 17 and standard deviation 1

E Mean 17 and standard deviation 5

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Minimum Increment operations to make Array unique

The UNIQUE function will return an array, which will spill if it's the final result of a formula. This means that Excel will dynamically create the appropriate sized array range when you press ENTER .

ArrayList is like an array behind the scenes as it’s backed by an array of type java.lang.Object in java.

Normally arrays are fixed in size which means the number of elements that can be added are fixed and cannot be increased once created. ArrayList can dynamically increase in size once the array that backs is full so, these way we won’t encounter runtime exceptions.

When we say list.get(0), it’s equivalent to array[0]. It can be said the ArrayList is a kind of more convenient way to use dynamic arrays.

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It appears that someone else claimed your dependant, assuming you submitted your dependent's details accurately. If another individual claim your dependent before you do, the IRS will reject your return since it processes the first return it gets. In such case, you must take the following actions:

1. File a paper return.

2. Document your case.

3. Answer when the IRS contacts you.

The IRS will reject a tax return filed electronically by a parent claiming a kid who has already been claimed for the year. The return will appear to have never been submitted. The dependent's tax ID number must be included on any future tax returns for the same tax year that are submitted electronically.

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The value of x + y is 5.

What is linear equation?

A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the preceding equation are y and x, and it is occasionally referred to as a "linear equation of two variables."

Given linear equations are

2x + 3y = 12  ....(i)

2x - y = 4    ......(ii)

To solve the give linear equation, using elimination method.

Subtract equation (ii) from equation (i)

2x + 3y = 12

2x - y = 4

(-)  (+)   (-)

________

   4y = 8

Divide both sides by 4:

y = 2

Putting y = 2 in equation (ii)

2x - 2 = 4

Add 2 with both sides:

2x = 6

Divide both sides by 2

x = 3

Putting x = 3 and y =2 in x + y:

x + y = 3 + 2 =5

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To find the x-intercept of a rational function we put y=0 in the given and find the corresponding value of x.To find the y-intercept of a  rational function we put x=0 in the function and the value of y.

The x-intercept is the point where a line crosses the x-axis.At this point y=0 and the y-intercepts is the point where a line crosses the y-axis.At this point x=0

It is point where the graph of the rational function intersects the x-axis or y-axis.

For Horzontal asymptotes

The horizontal asmptotes of a rational function can be determined by looking at the degree of the numerator and denominator.if Nis the degree of numerator and Dis the degree of denominator.Then N<D

So horizontal asymtote is y=0

To finding horizontal asymptotes we can just focus on degree of numerator and denominator.

For vertical asymptotes

vertical asymptotes occur where the denominator of a rational function approaches zero.A rational function cannot cross a vertical asymptotes because it would be dividing by zero. we set denomintor to zero and solve value of x.The values of x are vertical asymptotes.

Hence we can find easily horizontal and vertical asymptotes of rational function.

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Answer:

$1000

Step-by-step explanation:

5% = 1 year

50% = 10 years

50% = 0.5

Take 2000 times 0.5 = $1000

So, you will earn $1000 interest in 10 years

Answer:

Step-by-step explanation:

Grandpa Ernie has been shrinking for the past four years. His height decreased by a total of 2.4 cm during this period, which means that it went down by 0.6 cm each year. This gradual decrease in his height is quite remarkable given his age and overall health condition; however, it is not an entirely unexpected phenomenon due to aging and the natural decline of muscle mass with age as well as other factors such as poor nutrition or lack of exercise can contribute to this process too.

Answer:

You can tell if a function is positive or negative by checking the Y values. If all the Y values are above the x-axis it should be positive, and if the all y values are below the x-axis the function should be negative.

Good luck!

The factorized expression of (2a³ - 8a) ÷ 2a (a - 2) is (a - 4) ÷ (a - 2)

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

(2a^3-8a)÷ 2a (a-2)

Rewrite the above expression properly

So, we have the following representation

(2a³ - 8a) ÷ 2a (a - 2)

Divide the fraction by 21

This gives

(a - 4) ÷ (a - 2)

The above expression cannot be solved further

Hence, the solution is (a - 4) ÷ (a - 2)

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The coeficient of y² in the expression is the number 3

An algebraic expression is a set of numbers and letters that make up an expression that has a meaning, the letters are variables and the numbers are coefficients or independent terms, algebraic expressions can be part of an equation and model mathematical processes.

In the given expression we have the following:

3y² + 4x

 As we are asked for the coefficient of the variable "y" then we must take the number that is just before the letter which in this case is the number 3.

3 is the coefficient of y²

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Applying the area of a rectangle, the width of the walkway (x) is: 7 feet.

What is the Area of a Rectangle?

Area of a rectangle = (length)(width)

Total area of the pool and walkway = 621 sq. ft

Length of the pool and walkway = 13 + 2x

Width of the pool and walkway = 9 + 2x

Thus, we will have the equation of the total area as:

(13 + 2x)(9 + 2x) = 300

4x² + 44x + 117 = 621

4x² + 44x + 117 - 621 = 0

4x² + 44x - 504 = 0

Factorize

(x - 7)(x+18) = 0

x = 7

or

x = -18

The width, x, cannot be negative, so, the width of the walkway (x) would be 7 feet.

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Answer: [tex]0=\frac{2}{3}(x-3)(x-3)[/tex]

Step-by-step explanation:

The integration is as follow

Finding an antiderivative of a function is the process of integration. If a function can be integrated and its integral across the domain is finite with the given bounds, then the integration is definite.

Given:

first, [tex]\int\limits {e^{4x} / 19 - e^{8x}} \, dx[/tex]

= [tex]\int\limits {e^{4x} / 19 - e^{(4x)}^2} \, dx[/tex]

let [tex]e^{4x}[/tex] = z

4[tex]e^{4x}[/tex] dx = dz

= 1/4 [tex]\int\limits {dz / \sqrt{19} ^2 -z^2} \,[/tex]

= 1/4 x 1/2√19 log|(z+ √19)/(z-√19)| + C

= 1/8√19 log|([tex]e^{4x}[/tex] + √19)/ ([tex]e^{4x}[/tex] - √19)| + C

Second,

[tex]\int\limits {33 sec^5 (x)} \, dx[/tex]

= 33 [ 1/4 tan x sec³x + 3/4 ∫sec³x dx]

= 33/4  tan x sec³x + 99/4[ 1/2 tan x sec x +1/2 ∫sec x dx]

=  33/4  tan x sec³x + 99/8 tan x sec x +99/8 log (sec x+ tan x) + C

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Answer:

n - 6

Step-by-step explanation:

Number n is A less than B: n = B - A.

Answer:

I would say the length  is near 8

Answer:

In the given right angled triangle,

base(b)=2 mm and perpendicular(p)=7mm

hypotenuse (h)=?

By using Pythagoras theorem,

h²=p²+b²

or,h²=(7)²+(2)²

or,h²=49+4

or,h²=53

or,h=√53

or,h=7.28

The length of unknown side is 7.28 mm and it is not determined exactly.

The length 7.28 mm is between integers 7 and 8 and it is more closer to 7.

Answer:

Your answer is 2,600.019

Step-by-step explanation:

Very simply to answer it with or without a calculator.

2.6 * 10^3 + 0.19 * 10^-2

Without calculator
First move the decimal points before adding.

2.6 * 10^3 = 2,600.

0.19 * 10^-2 = 0.0019

This equation 2.6 * 10^3 + 0.19 * 10^-2 becomes

2,600 + 0.0019

Then add.

2,600.0019

Answer: 148cm²

Step-by-step explanation:

The volume of the cuboid is the length, height, and width multiplied together. Since the problem gives us the volume, we can set an equation.

[tex]l*w*h=120 cm^3[/tex]

Since the area of the base of 30cm², we can get another equation.

[tex]l*w=30cm^2[/tex]

We can plug in our equations.

[tex]30cm^2*h=120cm^3[/tex]          [divide both sides by 30cm²]

[tex]h=4cm[/tex]

Now, we know the height of the cuboid.

Since we aren't given a specific length and width, it can be any of the combinations that give 30 cm² when multiplied.

We can have 1,30 or 2,15 or 3,10 or 5,6.

I will work out 5,6 with l=5 and w=6.

The surface area of a cuboid is [tex]2(lw+wh+lh)[/tex].

[tex]2(30+24+20)=2(74)=148cm^2[/tex]

This is one possible answer. There are multiple ones with the pair choices above.

The height of Dubai fountain is 500 feet tall.

The foundational subject in mathematics is arithmetic, which covers operations with numbers. These include multiplication, division, addition, and subtraction. One of the key areas of mathematics, arithmetic serves as the cornerstone for students studying the topic of mathematics. The Greek term "arithmos," which means "numbers," is where the word "arithmetic" originates. Arithmetic is one of the oldest and most basic concepts in mathematics. It deals with numbers and the basic operations (addition, subtraction, multiplication, and division) that can be carried out on them.

Given

The Burj Khalifa in Dubai is the tallest building in the world, at 2,717 feet.

Burj Khalifa is 217 feet taller

Height of Dubai fountain

Subtracting

2717 - 217

2500

5 times the height

Divide by 5,

2500/5

500

The height of Dubai fountain is 500 feet tall.

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The value of polynomial for x = 2 that is f(2) will be 32.  

In many branches of mathematics and science, polynomials are used. For instance, they are used to define polynomial functions, which show up in settings ranging from basic chemistry and physics to economics and social science.

They are also used in calculus and numerical analysis to approximate other functions. Polynomial equations are used to form these functions. Polynomial rings and algebraic varieties, two fundamental ideas in algebra and algebraic geometry, are constructed in advanced mathematics using polynomials.

We have given polynomial f(x)=3x²8x+4

We have to find the value of f(2

We have to find the value of polynomial for x = 2

So for finding this value we have to just put x = 2 in polynomial

So f(2) will be

So the value of polynomial for x = 2 will be 32

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The equation 3x² - 8x - 15 is not factorizable.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

3x² - 8x  - 15

This is in the form of ax² + bx + c.

We will find the determinant D = b² - 4ac

If D = 0 or D = perfect square

The equation is factorizable.

Now,

a = 3

b = -8

c = -15

D = (-8)² - 4 x 3 x (-15)

D = 64 + 180

D = 244

This is not a zero nor a perfect square.

Thus,

3x² - 8x - 15 is not factorizable.

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The expression 3x+4y represents 3 bags of $3 tomatoes and 4 bags of $4 onions.

What is an expression?

The combination of numerical variables and operations expressed by the addition, subtraction, multiplication, and division signs constitutes a mathematical expression.

Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented using mathematical symbols. They can also be used to indicate other characteristics of logical grammar, such as the operation order.

Given that a basket of tomatoes costs $3, whereas an onion package costs $4. A customer buys y sacks of onions and x baskets of tomatoes.

Three bags of $3 tomatoes and four bags of $4 onions are denoted by the equation 3x+4y.

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The equation 3x² + 5x + 2 = 0 has two solutions, x = -2 and x = -1/3. This means that when the equation is set equal to zero and then solved, the two values of x that will make the equation true are -2 and -1/3.

To solve 3x² + 5x + 2 = 0, we must first find the roots of the equation. To do this, we can use the quadratic formula to solve for the two roots. The quadratic formula is x = [-b ± √(b² - 4ac)]/2a, where a, b, and c are the coefficients of the equation. To apply this to our equation, we have a = 3, b = 5, and c = 2, so the formula becomes x = [-5 ± √(5² - 4(3)(2))]/2(3). Simplifying this equation gives us x = [-5 ± √29]/6. The two solutions of this equation are x = -2 and x = -1/3. This means that when the equation is set equal to zero and then solved, the two values of x that will make the equation true are -2 and -1/3.

x = [-5 ± √29]/6

x = [-5 + √29]/6 = -2

x = [-5 - √29]/6 = -1/3

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Answer:

True False True False

Step-by-step explanation:

Independent and dependent variables are interdependent.

Dependent Variable: The term "dependent variable" refers to the variable that is being measured or tested in an experiment.

The results of the participants' tests, for instance, would be the dependent variable in a study looking at how tutoring affects test scores because that is what is being measured.

If you keep in mind that the dependent variable depends on the independent variable, finding it can be simpler.

This isn't the same as an experiment's independent variable, which is a variable that can stand on its own. The dependent variable (test results), however, may alter depending on the independent variable (tutoring).

The unrelated factor

Because the experimenters are allowed to change the independent variable as necessary, it is called a "independent" variable.

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The solution of the expression with three terms using binomial theorem can be solved by supposition and resubstituting the value.

Let us consider an example of three terms :

( 1 + x + x² ) ³

Use binomial theorem and expand the given expression:

Standard form of binomial expansion is :

( a + b )ⁿ = ⁿC₀aⁿb⁰ +  ⁿC₁aⁿ⁻¹b¹ + ⁿC₂aⁿ⁻²b² +..............+  ⁿCₙa⁰bⁿ

Let us consider x + x² = y

( 1 + y )³

=³C₀1³y⁰ +  ³C₁1³⁻¹y¹ + ³C₂1³⁻²y² +  ³C₃1⁰y³

= 1 + 3y + 3y² + y³

Resubstitute the value of y = x + x² we get,

= 1 + 3(x + x²) + 3(x + x²)² + (x + x²)³

= 1 + 3x + 3x² + 3( x² + 2x³ + x⁴) + ³C₀x³(x²)⁰ +  ³C₁x³⁻¹(x²)¹ + ³C₂(x)³⁻²(x²)² +  ³C₃x⁰(x²)³

= 1 + 3x + 3x² + 3x² + 6x³ + 3x⁴ + x³ + 3x⁴ + 3x⁵ + x⁶

= 1 + 3x + 6x² + 7x³ + 6x⁴ + 3x⁵ + x⁶

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