Why does -2^2= -4 but (-2)^2= 4?

Answer:

Leonhard Euler was a Swiss mathematician who lived from 1707-1783. He made significant contributions to many areas of mathematics, including number theory, calculus, and geometry.

One of Euler's significant contributions to the field of sequences and series is his work on the summation of infinite series. In particular, he developed a method for finding the sum of an infinite geometric series, which has the form:

a + ar + ar^2 + ar^3 + ...

where a is the first term, r is the common ratio, and the series continues infinitely.

Euler's method for finding the sum of this series is as follows:

If |r| < 1, then the sum of the series is:

a / (1 - r)

For example, let's consider the infinite geometric series:

2 + 4 + 8 + 16 + ...

In this series, a = 2 and r = 2. Since |r| < 1, we can use Euler's formula to find the sum:

sum = a / (1 - r)

= 2 / (1 - 2)

= -2

Therefore, the sum of the infinite geometric series 2 + 4 + 8 + 16 + ... is -2.

Euler's formula is just one example of his many contributions to mathematics. His work on sequences and series laid the foundation for many important concepts in calculus, and his ideas continue to influence the field of mathematics today.

By answering the presented question, we may conclude that Therefore, equation the two numbers are either 3 and 4, or 4 and 3.

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x Plus 3" equals the number "9." The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. The variable x is raised to the second power in the equation "x2 + 2x - 3 = 0." Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.

the system of equations:

[tex]x + y = 7\\x * y = 12 \\y = 7 - x\\x * (7 - x) = 12\\7x - x^2 = 12\\x^2 - 7x + 12 = 0\\(x - 3)(x - 4) = 0\\If x = 3, then y = 4\\If x = 4, then y = 3\\[/tex]

Therefore, the two numbers are either 3 and 4, or 4 and 3.

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Answer:

Step-by-step explanation:

The solution to the set of equations can be found by solving the system:

3x + 3y = 12

x + y = 4

We can simplify the second equation by solving for y:

y = 4 - x

Substituting this expression for y into the first equation, we get:

The solution to the set of equations can be found by solving the system:

3x + 3y = 12

x + y = 4

We can simplify the second equation by solving for y:

y = 4 - x

Substituting this expression for y into the first equation, we get:

3x + 3(4 - x) = 12

3x + 12 - 3x = 12

12 = 12

This is a true statement, which means that the system is consistent and has infinitely many solutions. Therefore, the correct answer is:

There are infinitely many solutions.

Answer:

Step-by-step explanation:

To find:-

Answer:-

We are here given that there are two linear equations, namely,

[tex]\begin{cases} 3x+3y = 12 \\ x+y = 4 \end{cases}[/tex]

These can be rewritten as ,

[tex]\begin{cases} 3x+3y - 12 =0\\ x+y -4=0 \end{cases}[/tex]

Before we precede we must know that,

Conditions for solvability :-

If there are two linear equations namely,

[tex]\begin{cases} a_x + b_1y+c_1 = 0 \\ a_2x+b_2y+c_2=0\end{cases}[/tex]

Then ,

Case 1 :-

If we have,

[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2} } \\[/tex]

Then , the lines are coincident and there are infinitely many solutions .

Case 2 :-

If we have,

[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2} } \\[/tex]

Then, the linear equations are inconsistent and have no solutions , thus the lines are parallel .

[tex]\rule{200}2[/tex]

So here with respect to angle standard form of pair linear equations, we have;

Hence here we have,

[tex]\longrightarrow \dfrac{a_1}{a_2} = \dfrac{3}{1} \\[/tex]

[tex]\longrightarrow \dfrac{b_1}{b_2}=\dfrac{3}{1} \\[/tex]

[tex]\longrightarrow \dfrac{c_1}{c_2}=\dfrac{-12}{-4}=\dfrac{3}{1} \\[/tex]

Therefore we can clearly see that,

[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2} =\boxed{\dfrac{3}{1}}} \\[/tex]

Hence there are infinitely many solutions and the lines are coincident .

Answer:

To find the slope of a line parallel to 3x - y = -1, we need to first rearrange this equation into slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Starting with 3x - y = -1, we can add y to both sides to get: 3x = y - 1 Then, we can add 1 to both sides to get: 3x + 1 = y This is now in slope-intercept form, with y = 3x + 1, so we can see that the slope of the line is 3. Since we want to find the slope of a line parallel to this one, we know that it must have the same slope of 3. Therefore, the answer is: Slope = 3.

Each student will read 2 1/2 pages aloud. The answer is option B, 2 1/2.

What is arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed constant number, called the common difference, to the preceding term.

For example, the sequence 2, 5, 8, 11, 14, ... is an arithmetic sequence with a common difference of 3, since each term after the first is found by adding 3 to the preceding term.

Ms. Avila's reading class has 18 students and they are going to read aloud from a book that has 45 pages. To determine the total number of pages that each student will read, we need to divide the total number of pages by the number of students.

So, 45 divided by 18 equals 2 1/2 pages per student.

This means that each student will read 2 1/2 pages aloud.

Therefore, the answer to the question is option B, 2 1/2.

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The measure of angle A is 59°

A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral. The circle which consists of all the vertices of any polygon on its circumference is known as the circumcircle or circumscribed circle.

The sum of opposite angle in a cyclic quadrilateral is 180. This means, 121+<A = 180°

therefore,

= angle A = 180-121

= 59°

therefore the measure of angle A is 59°

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we can conclude that the expression [tex]2x^2 + 3x -1[/tex] is the quadratic expression among the given options.

A quadratic expression is an algebraic expression in which the highest power of the variable is 2. The general form of a quadratic expression is:

[tex]ax^2 + bx + c[/tex]

where a, b, and c are constants and x is the variable. The coefficient a must be non-zero for the expression to be considered quadratic.

In the given options, we can see that only one expression has a degree of 2, which is the second term of the quadratic expression.

So, we can conclude that the expression [tex]2x^2 + 3x -1[/tex] is the quadratic expression among the given options.

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As a result, the range of the population's mean height for male Swedes is between 70.331 and 73.669 inches.

The range of a function or relation in mathematics is the collection of all feasible output values (also known as the dependent variable). It is the collection of all possible values that such a function can have. For instance, since its function can output anything non-negative real integer, f(x) = x2 would have a range of all non-negative real numbers. The range is frequently stated as a collection of numbers or as a circle, and it may be either finite or infinite.

given

For the population mean height of male Swedes, the following formula provides a 95% confidence interval:

where n is the sample size, x is the sample mean, y is the standard deviation of height, n* is the z-score corresponding to a 95% confidence level, which is 1.96, and z* is the z-score.

By entering the specified values, we obtain:

CI = 72 ± 1.96 * (3/√48)

CI = (70.331, 73.669) (70.331, 73.669)

As a result, the range of the population's mean height for male Swedes is between 70.331 and 73.669 inches.

By multiplying the critical value (1.96), by the standard error, one may determine the error bound:

EBM = 1.96 * (3/√48) ≈ 1.382

The error bound is therefore roughly 1.382 inches.

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Answer:

the area is 1000

Step-by-step explanation:

This one is for the boys with the booming system

Top down, AC with the cooler system

Answer:

10.5

Step-by-step explanation:

250% = 2.5

What number is 250% of 4.2?
We Take

2.5 x 4.2 = 10.5

So, 10.5 is 250% of 4.2

Answer: 5

Step-by-step explanation:

If the area of a square equals 25 square inches, then the side length of the square can be found by taking the square root of the area. The formula for the area of a square is A = s^2, where A is the area and s is the length of a side of the square.

So, in this case, we have:

A = 25 sq inches

s = sqrt(A) = sqrt(25 sq inches) = 5 inches

Therefore, the side of the square is 5 inches.

Answer:

a = b = 3√2

Step-by-step explanation:

Use trigonometry:

[tex] \sin(45°) = \frac{a}{6} [/tex]

Cross-multiply to find a:

[tex]a = 6 \times \sin(45°) = 6 \times \frac{ \sqrt{2} }{2} = \frac{6 \sqrt{2} }{2} = 3 \sqrt{2} [/tex]

Use the Pythagorean theorem to find b:

[tex] {b}^{2} = {6}^{2} - {a}^{2} [/tex]

[tex] {b}^{2} = {6}^{2} - ( {3 \sqrt{2}) }^{2} = 36 - 9 \times 2 = 36 - 18 = 18[/tex]

[tex]b > 0[/tex]

[tex]b = \sqrt{18} = 3 \sqrt{2} [/tex]

Answer: To find the margin of error and construct a 90% confidence interval for the population mean, we can use the following formula:

Margin of error = t-value x (standard deviation / square root of sample size)

where t-value is the value obtained from the t-distribution table with n-1 degrees of freedom and a confidence level of 90%.

For a sample size of 8 and a confidence level of 90%, the degrees of freedom is 7 and the t-value is 1.895 (obtained from a t-distribution table).

Using the given values, we can calculate the margin of error as:

Margin of error = 1.895 x (7.9 / sqrt(8)) ≈ 5.69 (rounded to one decimal place)

So the margin of error is approximately 5.69 miles.

To construct a 90% confidence interval for the population mean, we can use the formula:

Confidence interval = sample mean ± margin of error

Substituting the values, we get:

Confidence interval = 18.5 ± 5.69

So the 90% confidence interval for the population mean is (12.81, 24.19).

Interpretation:

We are 90% confident that the true mean driving distance to work for the population lies between 12.81 and 24.19 miles. This means that if we were to take many samples of 8 people and calculate the confidence interval for each sample, 90% of those intervals would contain the true population mean. The margin of error is the amount by which the sample mean may differ from the true population mean.

Step-by-step explanation:

Answer: We can simplify the expression as follows:

sin theta / (1 + sin theta) - sin theta / (1 - sin theta)

= [(sin theta)(1 - sin theta) - (sin theta)(1 + sin theta)] / [(1 + sin theta)(1 - sin theta)]

= [(sin theta - sin^2 theta) - (sin theta + sin^2 theta)] / [1 - sin^2 theta]

= -2 sin theta / cos^2 theta

= -2 tan theta

So the simplified expression is -2 tan theta, with no quotients.

Step-by-step explanation:

Answer:  (6, -2)

Step-by-step explanation:

         First, we will graph these equations. See attached. One has a y-intercept of -5 and then moves 2 units right for every unit up (we get this from the slope of 1/2). The other has a y-intercept of 4, and moves right one unit for every unit down (we get this from the slope of -1).

         The point of intersection is the solution, this is the point at which both graphed lines cross each other. Our solution is:

                         (6, -2)     x = 6, y = -2

The function f(2) from the piecewise function has a value of 0

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

The piecewise function

The x value of 2 belongs to the domain x > 2

So, we have

f(x) = -x + 2

Substitute the known values in the above equation, so, we have the following representation

f(2) = -2 + 2

Evaluate

f(2) = 0

Hence, the value iss 0

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The domain of F(d) is: {d ∈ R | d ≥ 0}

The domain is a mathematical concept that refers to the set of all possible input values (independent variables) for which a function is defined. In other words, the domain of a function is the range of values that can be used as input to the function.

Here we have

A large passenger airplane is making the 9,537-mile trip from New York to Singapore. During this flight, the plane will use 47,685 gallons of fuel or 5 gallons per mile of flight.

The function F(d) represents the amount of fuel, in gallons, consumed by the airplane on this trip after d miles of flight.

Therefore, we can express F(d) as:

F(d) = 5d

The domain of F(d) represents the set of all valid values of d for which the function is defined.

In this case, d represents the distance flown by airplane, and it must be a non-negative number since the plane cannot fly a negative distance.

Therefore,

The domain of F(d) is: {d ∈ R | d ≥ 0}

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Answer:

The triangles are similar by AA since vertical angles are congruent.

[tex] \frac{8}{12} = \frac{x}{7} [/tex]

[tex]12x = 56[/tex]

[tex]x = \frac{14}{3} = 4 \frac{2}{3} [/tex]

0.2525 is the probability that a simple random sample.

What is the simple definition of probability?

A probability is a number that expresses the possibility or likelihood that a specific event will take place. Probabilities can be stated as proportions with a range of 0 to 1, or as percentages with a range of 0% to 100%. Probability is a measure of how likely or likely-possible something is to happen.

Since the distribution of the life expectancies of a certain protozoan is normal, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = life expectancies of the certain protozoan.

µ = mean

σ = standard deviation

n = number of samples

From the information given,

µ = 46 days

σ = 10.5 days

n = 49

The probability that a simple random sample of 49 protozoa will have a mean life expectancy of 47 or more days is expressed as

P(x ≥ 47) = 1 - P(x < 47)

For x = 47

z = (47 - 46)/(10.5/√49) = 0.67

Looking at the normal distribution table, the probability corresponding to the z score is 0.0.75

P(x ≥ 47) = 1 - 0.75 = 0.25

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The Two figures are (Not similar) because 2/5 ( does not equal) 3/6.

What is similar rectangle?

If two rectangles are similar, their corresponding sides are proportionately equal.

Now comparing the length and width of the two rectangles then.

=> 2/5 = 3/6

Which is not equally proportional.

Hence the both rectangles are similar because 2/5 does not equal 3/6 .

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Answer:

Step-by-step explanation:

I am assuming the triangles are similar.

You can make a proportion between the two triangles and solve for x but you must match the sides properly.

RS ≈ MN

ST ≈ NO

Making a proportion between the sides:

[tex]\frac{x}{3}[/tex] = [tex]\frac{6}{4}[/tex]  

cross multiplyand then divide

6 · 3 = 18

18 ÷ 4 = 4.5

x = 4.5

If you check the ratios - they are all the same.

For example  6 ÷ 4 = 1.5

7.5 ÷ 5 = 1.5

4.5 ÷ 3 = 1.5

this checks out

The desalination plant can produce [tex]9.84*10^6[/tex] gallons of water in four day in scientific notation.

We must divide the daily production rate by the number of days in order to get the total volume of water that a desalination plant can generate in four days:

4. days at a rate of [tex]2.46*10^6[/tex] gallons equals [tex]9.84*10^6[/tex] gallons.

Hence, in four days, the desalination plant can produce [tex]9.84 * 10^6[/tex] gallons of water.

Large or small numbers can be conveniently represented using scientific notation, especially when working with measurements in science and engineering. A number is written in scientific notation as a coefficient multiplied by 10 and raised to a power of some exponent. For example, [tex]2.46*10^6[/tex] denotes 2,460,000, which is 2.46 multiplied by 10 to the power of 6.

The solution in this case is [tex]9.84*10^6[/tex], or 9,840,000, which is 9.84 multiplied by 10 to the power of 6. Large numbers can be written and compared more easily using this format, and scientific notation rules can be used to conduct computations with them.

In conclusion, the desalination plant has a four-day capacity of [tex]9.84 * 10^6[/tex] gallons of water production.

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Possible rational zeros of the polynomial is

±1, ±1/2, ±1/4, ±2, ±5, ±5/2, ±5/4, ±10.

Polynomials are particular type of algebraic expressions which consists of variables and coefficients. Various arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions can be performed on polynomial but not division by variable. An example of a polynomial is x-12 . Here it has two  terms:  x and -12.

Given polynomial p(x)= 4x⁵-2x³+10

The polynomial can be rewritten as p(x)= 4x⁵+ 0x⁴-2x³+0x²+0x+10

Comparing the polynomial with

p(x) = aₙ xⁿ + aₙ₋₁xⁿ⁻¹+ ---------- + a₁ x + a₀ we get the  leading coefficient

aₙ= 4 and the constant term a₀= 10

So the possible zeros are=  ±( factors of a₀)/ (factors of aₙ)

Factors of 10:

1, 2, 5, 10

Factors of 4:

1, 2, 4

Hence, Possible rational zeros:

±1, ±1/2, ±1/4, ±2, ±5, ±5/2, ±5/4, ±10

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Answer: 165 degrees

Step-by-step explanation:

Let's start off and name the angle between the hour and minute hand x, representing the number of degrees between the hands. Next, let's look at the minute hand first. After exactly an hour, the minute hand does not change its position(since after one hour, it comes back to where it was previously). As for the hour hand however, its position differs by exactly "1 hour tick", which is equivalent to 30 degrees (360 / 12 hours). What that implies is that x would have to equal x + 30, which is clearly impossible. However, we can think about the problem in a different way: by looking at the acute angle only between the hands, that would indeed be possible. Assuming that x + 30 is greater than 180(since if it wasn't, that wouldn't be possible), we can write the following equation:

x = 360 - (x+30)

, with the right hand side coming from the fact that the obtuse angle of x + 30 must have an acute counterpart, of which both add up to 360.

Then, we can simplify to:

x = 330 - x

or:

2x = 330

x = 165.

This gives us our only answer of 165 degrees

So, the corresponding point for g(x) on the graph of f(x) is (-3/2, F (-3/2)).

In mathematics, a function is a relation between a set of inputs (the function's domain) and a set of possible outputs (the function's range) with the property that each input is associated with exactly one output.

Functions are often described using a formula or an equation, such as [tex]f(x) = x^2,[/tex] where "f" is the name of the function, "x" is the input variable, and "[tex]x^2[/tex]" is the output. The value of the function for a specific input value of "x" can be found by substituting that value into the equation.

Functions are used in many areas of mathematics, science, engineering, and economics to model real-world phenomena, analyze data, and solve problems. They are also important in computer science, where they are used to write algorithms and design software.

if we know the point (x, y) on the graph of a function f(x), we can find the corresponding point for the function g(x) = F(-1/2x) by substituting -1/2x for x in the expression for f(x) and simplifying. That is, we can find the point (u, v) on the graph of g(x) such that u = -1/2x and v = F(-1/2x), where (x, y) is a point on the graph of f(x).

For example, if f(x) = x^2 and the point (3, 9) is on the graph of f(x), then we can find the corresponding point for g(x) = F(-1/2x) by substituting -1/2x for x in the expression for f(x) and simplifying:

[tex]u = -1/2x = -1/2(3) = -3/2\\v = F(-1/2x) = F(-1/2(3)) = F(-3/2)[/tex]

So, the corresponding point for g(x) on the graph of f(x) is (-3/2, F(-3/2)). However, without knowing the specific definition of fix,

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The graph that shows the solution of the inequality 7 ≥ n + 5 is the graph of option D.

What is the solution to the inequality 7 ≥ n + 5?

The solution to the inequality 7 ≥ n + 5 is as follows:

7 ≥ n + 5

7 - 5 ≥ n

2 ≥ n

n ≤ 2

2, Let p be the number of pages Eduardo wrote before today. Then, the inequality that represents the situation is:

p + 16 > 50

To solve for p, first subtract 16 from both sides of the inequality:

p + 16 - 16 > 50 - 16

p > 34

Therefore, Eduardo wrote at least 34 pages before today.

Let r be the number of rows of seeds the farmer needs to plant. Then, the equation that represents the situation is:

6.5r ≥ 52

divide both sides of the equation by 6.5:

r ≥ 8

Therefore, the farmer needs to plant at least 8 rows of seeds to harvest at least 52 bushels of tomatoes.

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Answer: Given a 2x2 matrix A:

A = [a11 a12]

   [a21 a22]

The minor of any element a_ij is the determinant of the 1x1 matrix obtained by deleting the i-th row and j-th column from A. In other words, the minor of a_ij is given by:

M_ij = det(A_ij)

where A_ij is the matrix obtained by deleting the i-th row and j-th column from A.

For example, the minor of a11 is given by:

M_11 = det([a22])

    = a22

Similarly, we can find the minors of the other elements:

M_12 = det([a21])

    = a21

M_21 = det([a12])

    = a12

M_22 = det([a11])

    = a11

Therefore, the minors of the matrix A are:

M = [a22  a21]

   [a12  a11]

Step-by-step explanation:

Answer:

[tex](x+1)^{2} +(y+2)^{2} =20[/tex]

Step-by-step explanation:

Using the mid point formula to find the center of the circle:

midpoint = [tex](\frac{x_{1} +x_{2} }{2} ,\frac{y_{1} +y_{2} }{2} )[/tex]

Midpoint = [tex](\frac{3+-5}{2} ,\frac{0+-4}{2} )[/tex]

Midpoint = [tex](-1,-2)[/tex]

The midpoint is the same as the centre of the circle

Find the distance(the diameter of the circle) between those two points to find the radius:

Distance formula = [tex]\sqrt{(y_{2}-y_{1} )^{2} +(x_{2} -x_{1} ) ^{2} }[/tex]

Distance formula = [tex]\sqrt{(-4-0)^{2} +(-5-3)^{2} }[/tex]

Distance formula = [tex]\sqrt{16+64}[/tex]

Distance formula = [tex]\sqrt{80}[/tex]

So,the diameter is [tex]\sqrt{80}[/tex] and to find the radius we need to divide the diameter by two

Radius = [tex]\frac{\sqrt{80} }{2}[/tex]

Radius = [tex]2\sqrt{5}[/tex]

the equation of circle:

Radius = [tex]2\sqrt{5}[/tex]

Center = (-1,-2)

[tex](x-h)^{2} +(y-k)^{2} =r^{2}[/tex]

[tex](x- -1)^{2} +(y- - 2)^{2} =(2\sqrt{5} )^{2}[/tex]

[tex](x+1)^{2} +(y+2)^{2} =20[/tex]