What additional information do you need to know in order to prove the two triangles congruent using HL?

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.

The total cost for the meal with tax and tip is $22.26.

This can be calculated using the following formula:

Total Cost with Tax and Tip = Meal Cost + (Meal Cost * Sales Tax Rate) + (Meal Cost * Tip Rate)

In this case, the formula would be:

Total Cost with Tax and Tip = 17.00 + (17.00 * 0.07) + (17.00 * 0.18)

This simplifies to:

Total Cost with Tax and Tip = 17.00 + 1.19 + 3.06

Which yields a final total cost of 22.26.

To explain in more detail, the meal cost is the original cost of the meal, which in this case is 17.00. Then, the sales tax rate is applied to the meal cost, which is 7% or 0.07. This gives us an additional cost of 1.19 (17.00 * 0.07). Lastly, the tip rate is applied to the meal cost, which is 18% or 0.18. This gives us an additional cost of 3.06 (17.00 * 0.18). Adding the meal cost of 17.00, the tax cost of 1.19, and the tip cost of 3.06 together gives us a total cost of 22.26.

Therefore, the total cost for the meal with tax and tip is $22.26.

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3 to 1 slope means, there are three horizontal units for every one vertical unit.

The slope is computed mathematically as "rise over run" (change in y divided by change in x).

In accordance with the slope equation, the slope of such a line may be determined by dividing the rise of the line between two positions by the run of the same line between the same two sites.

That is to say,

"Ratios like 3:1 are used to represent slopes. This indicates that there is a 1 unit (foot or meter) vertical shift, either up or down, for every 3 units (feet or meters) of horizontal distance."

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The statistical mode of a set of numbers is the number that occurs most frequently in the set. In the set {9, 10, 6, 5, 6}, the number 6 occurs twice, which is more than any other number. Therefore, the mode of this set is 6.

The field of statistics, which is applied mathematics, encompasses the gathering, describing, analyzing, and drawing of conclusions from quantitative data. The fundamental mathematical ideas in statistics include calculus of differential and integrals, linear algebra, and probability theory.

When attempting to estimate the population of a town, it is vital to determine the percentage of its residents who watch TV. This context makes reference to the sample, which is a condensed selection from the population.

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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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Therefore , the given expression of the problem √2 = 2 is a false statement the true statement is √2 ≠ 2.

The number is derived by multiplying the square root of a positive integer by itself.

For instance, the square root of 25 is 5, since 5 x 5 is 25.

As you can see, 5 is also the square root of 25 because (5) (5) = 25.

The square root of a number is always represented by the symbol, which is √.

Here,

The square root symbol, which has the value of 1.414, is used to symbolize the square root of 2, also known as root 2.

Mathematicians frequently utilize this number.

Because Root 2 has an unlimited number of decimals and cannot be stated as a fraction, it is an irrational number.

and 2 has a value of 2.

Thus,

=> √2 ≠ 2

as

=> 1.414 ≠ 2.00

Therefore , the given expression of the problem √2 = 2 is a false statement the true statement is √2 ≠ 2.

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The 12 types of sets are empty set, singleton set, finite set, infinite set, subset, equal set, universal set, power set, proper subset, superset, equivalent set and proper superset.

An empty set does not contain any element, while a singleton set contain only one element.

If the number of elements in  a set is finite, the set is a finite set. If the number of elements in  a set is not finite, that is not countable it is an infinite set.

A set, say A contain only elements of another set, say B but not necessarily all elements, than set A is a subset of set B.

Whenever a set A is a subset of set B, we say the B is a superset of A. If A and B are two sets, then A is called the proper subset of B if A is a subset of B but B is not a subset of A i.e., A ≠ B.

If a set contains all and only elements of another set, then both set are said to be equal sets.

A universal set contains all elements of related sets.

Power set is the collection of all subsets of a particular set.

The set B is the proper superset of set A, then all the elements of set A are in B, but set B must contain at least one element which is not present in set A.

Equivalent sets are sets that have same number of elements.

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The required mean of the given data is 5.71.

The sum of all values divided by the total number of values determines the mean (also known as the arithmetic mean, which differs from the geometric mean) of a dataset.

The term "average" is frequently used to describe this measure of central tendency.

So, we have the data set:

2, 2, 4, 6, 8, 8, 10

Mean formula: Sum of terms / Number of terms

Now, calculate the mean as follows:

Mean: Sum of terms / Number of terms

Mean: 40 / 7

Mean: 5.71

Therefore, the required mean of the given data is 5.71.

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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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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: 2% is equivalent to 1/50

Step-by-step explanation:

We first need to divide the numerator by the denominator.

1 ÷ 50 = 0.02

Once we have the fraction in a decimal format, the answer is then multiplied by 100 to get the correct percentage:

0.02 × 100 = 2%

Let x= 3^4. So the value of given expression after applying power rule is x=81.

What is meant by power?

How many times you should multiply a number depends on its power. Exponents and indices are other names for powers. For instance, 82 may also be referred to as "8 to the power 2," "8 to the second power," or just "8 squared."

What is meant by n^5 where n can be any number?

By n^5 means n^5=n*n*n*n*n that is multiplying n five times.

3^4=(3^2)^2

=(9)^2

=81

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Answer:

Step-by-step explanation:

value of y is 87*

because linear pair always sum up to give 180*

we know 93+y=180

so y=180-93=87

y= 87°

180-93 = 87

The fraction of the instructions that used the word off is [tex]\frac{9}{25}[/tex]

Generally it is not possible. But if someone applying for early retirement after being forced to leave their jobs due to an illness or injury. In order to help them pay their costs until their SSDI claim is approved and the waiting period for those benefits has passed, they can start collecting early retirement payments.

Generally speaking, it is not possible to receive both Social Security retirement benefits and Social Security Disability Insurance (SSDI) at the same time. However, if you fulfill the severe financial requirements while receiving SSDI or retirement payments from Social Security, you can be eligible for Supplemental Security Income (SSI).

To fill the gap between when someone must leave the working due to a disability and when they can receive retirement benefits, the Social Security Administration (SSA) established the SSDI program. Because of this, there is only one option to get SSDI and retirement benefits at the same time.

This happens when a person chooses to retire early between the ages of 62 and their full retirement age but is later granted SSDI payments. Some people do this on purpose by applying for early retirement after being forced to leave their jobs due to an illness or injury. In order to help them pay their costs until their SSDI claim is approved and the waiting period for those benefits has passed, they can start collecting early retirement payments.

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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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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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Based on the given task content; the percent chance is 1 out of 20 is 5%.

Percentage refers to the amount, number or rate of something, regarded as part of a total of 100.

The percent chance is 1 out of 20

= 1/20 × 100

= 0.05 × 100

= 5%

Un conclusion, 5% is the percent chance is 1 out of 20

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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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(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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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.

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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Answer:

Substitution method = plug in the slope and the (x, y) point values into y = mx + b, then solve for b. ...

Point-slope form = y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the point given and m is the slope given.

Step-by-step explanation:

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:

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.

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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Pascal's law statement states that pressure is equal to the force divided by the area on which it acts.

According to Pascal's law, pressure applied to a fluid inside of a container will be transmitted without a change in magnitude to every point within the fluid as well as the container's walls. Any point in the fluid experiences equal pressure in all directions.

The force multiplied by the surface area on which it acts produces pressure. Pascal's principle states that a pressure increase on one piston in a hydraulic system causes an equal increase in pressure on another piston in the system.

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