ryan planted flowers in a rectangular garden. the width of the garden is (x-7) feet and the length of the garden is 3 feet longer than the width. write an expression for the area of the garden.

The quotient and remainder are x³ -5x² + 10x - 17 and 55.

What is Synthetic division?

When the divisor is a linear factor, synthetic division is a technique used to carry out the division operation on polynomials.

Here, (x+2) is a linear factor which indicates that synthetic division can be applied.

So, we will divide the x^4 - 3x^3 - 7x + 1 by x+2.

(refer the attached solution)

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Depending on the level of difficulty, there are three different dividing methods. These include the bus stop approach, the chunking method, and the long division method. The chunking method is also known as division by repeated subtraction.

One of the four fundamental arithmetic operations, or the process by which two or more numbers are added together to create a new number, is division. Multiplication, addition, and subtraction make up the remaining operations.

The opposite of multiplication is division.

Depending on the level of difficulty, there are three different dividing methods. These include the bus stop approach, the chunking method, and the long division.

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The root 4 is called quad root.

When a number is raised to the root 4, then it is called quad root of that number as the term quad in Latin refers to 4, just the way when a number is raised to the root 2 it is called square root of that number, when a number is raised to the root 3, it is called cube root of that number.

For example, the quad root of the number 256 is (256) ^1/4 which results to 4 as the answer since 4 x 4 x 4 x 4 = 256.

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Answer:

800 m^2

Step-by-step explanation:

= 1250 - 450

= 800 m^2 est la superficie de la terre de M. Vergne

j'espère que ma réponse t'aidera.

Answer:

9 and 11

Step-by-step explanation:

there is a difference of 2 between consecutive odd integers.

let n and n + 2 be the integers , then

n + n + 2 = 20

2n + 2 = 20 ( subtract 2 from both sides )

2n = 18 ( divide both sides by 2 )

n = 9

and n + 2 = 9 + 2 = 11

the two consecutive odd numbers are 9 and 11

The expression represents the number of coins Arthur is x + 80 cents.

The term expression in math refers a sentence with a minimum of two numbers or variables and at least one math operation.

Here we have given that Arthur has x pennies. His father gave him 6 dimes, and his mother gave him 4 nickels.

And we need to find expression represents the number of coins Arthur has now.

From the given question, here we have to convert the pennies, dimes and nickels into same unit before adding to get the expression for the number of coins Arthur will have in total.

AS we know that one penny = 1 cent, one dime = 10 cents, one nickel = 5 cents and

Therefore, x pennies = x cents

Then 6 dimes = 60cents

Then 4 nickels = 20 cents

Therefore, the Arthur's coins in total is written as,

=> (x + 60 + 40) cents

=> (x + 80) cents.

Therefore, the resulting expression is written as (x + 80) cents.

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The sum of the squares of three consecutive odd number   11, 13, 15 is 515.

What are consecutive numbers ?

Numbers that follow each other continuously in the order from smallest to largest are called consecutive numbers. For example: 1, 2, 3, 4, 5, 6, and so on are consecutive numbers.

Let the second number = x.

so the first number = x-2

the third number = x + 2

according to the question, the sum of the squares of the numbers = 515

so the formula is:

x² + (x-2)² + (x + 2)² = 515

(a-b)² = a² + b² - 2ab

and,

(a+b)² = a² + b² + 2ab

So, let’s apply these to our equation:

x² + x² + 2² -2*2x + x² + 2² + 2*2x = 515

There are three x² and 2*2 = 4 and 2*2x = 4x. Let’s apply all that to the equation,

3x² + 4 - 4x + 4 + 4x = 515

there is a -4x and a +4x, -4x + 4x = 0, so let’s remove that.

3x² + 4 + 4 = 515

3x² + 8 = 515

3x² = 515 - 8

3x² = 507

x² = 507/3

x² = 169

x = √169

√169 (square root(169)) = 13

x = 13

if x = 13, then the first number = 13–2 = 11 and the third number = 13 + 2 = 15

Hence , the sum of the squares of three consecutive odd number   11, 13, 15 is 515.

11*11 + 13*13 + 15*15 = 515

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Answer:

Step-by-step explanation:

You want to know how the form of a linear equation can be used to reveal its key characteristics.

The characteristics of a linear equation that are of interest for this problem are ...

A linear equation can be written in many forms. Some of these forms make it easy to determine one or more of the key characteristic listed above. Some of the more common forms are Slope-Intercept form, Point-Slope form, and Standard form. There are a number of others.

The slope-intercept form is ...

  y = mx + b

where m is the slope, and b is the y-intercept. This form makes it easy to determine ...

This equation cannot be used to describe a vertical line.

We can rearrange the above equation to reveal the x-intercept.

  y = m(x -a)

where m is the slope and 'a' is the x-intercept. This form makes it easy to determine ...

This equation cannot be used to describe a horizontal line (except y=0).

The equation of a line with slope m through point (h, k) can be written as ...

  y -k = m(x -h)

This makes it easy to determine ...

This equation cannot be used to describe a vertical line.

The standard form equation of a line is similar in style to the standard form equations for other geometric figures (circle, ellipse, hyperbola). This form is ...

  Ax +By = C

where A, B, C are mutually prime (integers), and A ≥ 0. If A = 0, then B > 0. That is, the leading coefficient is positive. Integers are preferred for the coefficients, but some lines have irrational slope or other key characteristics, so integers may not be able to be used in all places.

The key characteristics cannot be read from the equation directly, but they are one step away:

Solving for the intercepts is especially easy, so we list this form as one that can be efficient for determining the intercepts.

A very nice form for efficiently determining the intercepts, or efficiently writing an equation from the intercepts is ...

  x/a +y/b = 1

where 'a' is the x-intercept, and 'b' is the y-intercept. This form makes it very easy to determine ...

You may notice that dividing the standard form equation by C and rearranging the coefficients will give you this form:

  Ax +By = C

  (A/C)x +(B/C)y = 1 . . . . . . . divide by C

  x/(C/A) +y/(C/B) = 1 . . . . . . . intercept form

The general form of a polynomial equation of any degree is ...

  f(x, y) = 0

The general form of a linear equation is ...

  Ax +By +C = 0

This is simply a rearrangement of the Standard form equation. Equations written in this form can be helpful for ...

When two points are given, and you want an equation of the line, various forms are available for using those coordinates. These forms make use of the fact that the slope of a line is the same everywhere, so the ratio of any difference of y-coordinates to the corresponding difference of x-coordinates is a constant (the slope). Again, slopes of 0 or "undefined" can cause trouble with certain of these forms. Here are several different arrangements of 2-point forms:

  [tex]y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)\\\\(x_2-x_1)(y-y_1)=(y_2-y_1)(x-x_1)\\\\\dfrac{y-y_1}{x-x_1}=\dfrac{y_2-y_1}{x_2-x_1}=m\qquad\text{Slope form}\\\\\\\dfrac{x-x_1}{y-y_1}=\dfrac{x_2-x_1}{y_2-y_1}\qquad\text{inverted from the above}\\\\\\\dfrac{y-y_1}{y_2-y_1}=\dfrac{x-x_1}{x_2-x_1}[/tex]

The last three "ratio forms" are variations of each other. Any such proportion can be written 4 ways. (3 are shown) The form not shown has the variables in the denominator, which is usually not useful.

All of these forms can be written in a "general form" by subtracting one side.

The slope can be read directly from the 1st and 3rd of the forms shown here. It is m = (y2-y1)/(x2-x1).

The slope form is useful for determining ...

Another 2-point form is one written as a determinant:

  [tex]\left|\begin{array}{cc}x-x_1&x_2-x_1\\y-y_1&y_2-y_1\end{array}\right|=0[/tex]

__

Additional comment

It can be useful to recognize the various forms and their relationship to known points on the line or other key characteristics of the function. Any of the slope or point-slope forms make it easy to write the equations of parallel or perpendicular lines through a given point. (The standard and general forms can be used for this, too.)

The value of the side is needed to verify the SAS rule.

In triangle theorem SAS states that if two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle then the two triangles are congruent.

Here we have to find the additional information needed to enable us to apply the SAS postulate in the given triangle.

Let us consider the following triangle, through the given triangle, we must know that in order to the SAS Postulate, here you must need two sides and the included angle in both triangles.

Therefore, here you need the side on the other side of the angle.

So, while we looking into the given triangle, we know that the △ABC, has the side BC and in △LKM that is KM.

Therefore, if it is congruent, when

=>BC ≅KM

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Answer:

The average rate of change is -4

Step-by-step explanation:

The given function is  f(x) = -(3)x1+3

We want to find the average rate of exchange of this function from x=1 to x=3.

This is the slope of the secant line joining the points (1,f(1)) and (3,f(3)).

From the graph, f(1)=2 and f(3)= -6

The average rate of exchange

=f(3)-f(1)/3-1

=-6-2/3-1

=-8/2

=-4

When an initial/input value is zero, the output of a function is its initial value.

The starting value represents the input at zero, whereas the rate of change relates to how rapidly the output varies in respect to the input.

Thus, if x = 0 in a function, the initial value is defined. It serves as the scenario's beginning point in real life.

A linear equation's y-intercept is frequently thought of as initial value of the function.

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The Total Cost that people must pay to rent x scooters is given by the equation y = 80.5x

What is Linear Equation in Two Variables?

A linear equation in two variables is one that is stated in the form ax + by + c = 0, where a, b, and c are real integers and the coefficients of x and y, i.e. a and b, are not equal to zero.

Solution:

Total Cost = y

Number of Scooters Rented = x

y = (20.5$ + 7.5*8 $)x

y = (20.5 + 60)x

y = 80.5x

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Answer:

The scale factor is 60

Step-by-step explanation:

The drawing is measured in inches and the billboard is measured in feet.  First, convert 15 feet to 180 inches.  Then use the ratio [tex]\frac{180}{3}[/tex] to find that the scale factor is 60.  This means the length of the billboard will be 600 inches or 50 feet.

Answer: x^2 + y^2 + 18y + 81 = 0

Step-by-step explanation:

The equation of a circle with center at the point (0, -9) and radius 9 is given by the following equation:

(x - 0)^2 + (y + 9)^2 = 9^2

This can be simplified to the following:

x^2 + y^2 + 18y + 81 = 0

This is the equation of a circle with center at (0, -9) and radius 9.

Answer:

  1/6 pan

Step-by-step explanation:

You want to know the portion of the whole represented by 1/4 of 2/3.

Todd ate 1/4 of 2/3 of the pan of brownies. That fraction of the original whole amount is ...

  (1/4)×(2/3) = (1·2)/(4·3) = 2/12 = 1/6

Todd ate 1/6 of the entire pan.

Answer:

[tex]\frac{5}{36} , \frac{6}{49}[/tex]

Step-by-step explanation:

The numerator of these terms is increasing by 1

The denominator is being squared... Look at this

The denominator of the first term is 2² which equals 4

the next denominator is 3² = 9

the next 4² = 16

the next 5² = 25

so, what what's next?

6² = 36   and 7² = 49

Hope I helped! :)

Answer:

The next two terms of the following sequence is 5/36 and 6/49.

Step-by-step explanation:

The pattern in this sequence is that the numerators of the sequence are all positive integers that increase by 1, so I added 1 to each preceding numerator. The denominators of the sequence are consecutive perfect squares (e.g., 4, 9, 16, 25, 36, etc.), so I made each following denominator the next perfect square.

The criterion if any could prove the triangles are congruent are SSS, SAS, ASA, AAS, and RHS congruence properties.

Two triangles are congruent if all the three corresponding sides of two triangles are the same. There are a few more criteria also. Thus, congruent triangles are mirror images to each other. The mathematical symbol represents congruence is ≅.

In the triangles ΔABC, ΔEFG as shown below, we can identify corresponding angles and corresponding sides are equal.

Diagram attached at the end of solution.

ΔABC≅ΔEFG

Corresponding Angles: ∠A≅∠E,∠B≅∠F,∠C≅∠G

Corresponding sides: AB≅EF,BC≅FG,AC≅EG.

Corresponding vertices: A and E,B and F,C and G

Two triangles of the same size and shape are called congruent triangles. When we rotate, reflect, or translate a triangle, its position or appearance seems identical to the other, also called congruent.

Based on the experiments, there are mainly 5 conditions or rules to compare the two triangles to be congruent. They are SSS, SAS, ASA, AAS, and RHS congruence properties.

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Answer: t 13/3 ≈0.230769231

Step-by-step explanation:

Answer:

(b) 91

Step-by-step explanation:

The probability of an item scanning correctly is 98.9%, so the probability of an item scanning incorrectly is 1 - 98.9% = 1.1%.

The expected number of items to buy to find one that scans incorrectly is 1 / 1.1% = 91. This means that you would expect to buy 91 items on average to find one that scans incorrectly.

Rounding up to the next integer, the answer is 92. Therefore, the correct answer is (b) 91.

The size and shape of the object or pre-image are not changed by a rigid transformation, sometimes referred to as isometry, when the final picture is returned.

While the image's size and shape are unaffected by rigid body motion, its location and orientation are. The three fundamental movements of a rigid body are translation, reflection, and rotation. Prior to movement, archetypes depict points or forms.

The size and shape of the object or pre-image are not changed by a rigid transformation, sometimes referred to as isometry, when the final picture is returned. Rigid transformations are any of the three recognized transformations, which include rotation, reflection, and translation.

Triangle = 1/2 X b X h

b= 10

h=5

A = 1/2 X 10X 5

A = 25 unit sq.

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It is said that concept of using 0 numerically came from India. It is said that Aryabhatta used zero as a placeholder number in the 5th century and thus said to invent 0. There are various theories around the invention of 0.

Therefore, the answer is Aryabhatta.

Zero as a symbol and as absence of quantity helps us in calculations. As a placeholder various civilizations have utilized zero, but they didn't develop the idea of zero as a number. In the 7th century, Brahmagupta have noted laws on use of zero numerically.

Zero as a number represents an empty quantity. It is denoted by the arithmetic symbol 0.

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Answer:

No, she is wrong

Step-by-step explanation:

(4x - 6)^2 can be write as

(4x -6) ( 4x-6)

16x^2 - 24x -24x + 36

16x^2 -48x + 36

So, she is wrong!

Hey there!

(4x - 6)^2

= (4x - 6)(4x - 6)

FACTOR/DISTRIBUTE

= 4x(4x) + 4x(-6) - 6(4x) - 6(-6)

SIMPLIFY

= 16x^2 - 24x - 24x - 36

COMBINE the LIKE TERMS

= 16x^2 - 48x - 36

Therefore, the answer should be:

16x^2 - 48x - 36

Therefore, your answer should be: “No, because the equation should’ve been 16x^2 - 48x - 36. She didn’t factor the equation correctly”

Good luck with your assignment & enjoy your day!

~Amphitrite1040:)

The set of integers is closed under the operation of addition because the sum of any two integers is always another integer and is therefore in the set of integers.

What are integers ?

An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples of integers are: -5, 1, 5, 8, 97, and 3,043. Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14, .

According to the Property of Closure, A set has the closure property under a particular operation if the result of the operation is always an element in the set.  If a set has the closure property under a particular operation, then we say that the set is “closed under the operation.”

Hence ,  the set of integers is closed under the operation of addition because the sum of any two integers is always another integer and is therefore in the set of integers.

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Since there is a common ratio between each term so this is a geometric sequence.

A unique kind of sequence is a geometric sequence. Every term in the series (aside from the first term) is multiplied by a fixed amount to determine the following term. In other words, we multiply the current phrase in the geometric sequence by a constant term (called the common ratio), and then divide the current term in the geometric sequence by the same common ratio to discover the previous term in the geometric sequence.

Every term in a geometric series is obtained by multiplying the term before it by the same number. [tex]a_{n}=a_{1} r^{n-1}[/tex] is the general phrase for it. The common ratio is denoted by the number r. Any term in the sequence can be used to find it by dividing it by the word before it.

In 72 36 and 18 if we multiply the previous term by [tex]\frac{1}{2}[/tex] we will get the next term. So r=[tex]\frac{1}{2}[/tex] and hence there is a common ratio so the given sequence is geometric sequence.

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Answer:

-15/4

Step-by-step explanation:

m=(y2-y1)/(x2-x1)

  =(3-18)/(-14- -18)

  =-15/4

Answer:

Slope = [tex] \sf \frac{-15}{4} [/tex]

Step-by-step explanation:

Now we have to,

→ find the required slope.

Formula we use,

→ Slope(m) = (y2 - y1)/(x2 - x1)

Then the slope will be,

→ m = (y2 - y1)/(x2 - x1)

→ m = (3 - 18)/(-14 -(-18))

→ m = -15/(-14 + 18)

→ [ m = -15/4 ]

Hence, the slope is -15/4.

Positive Slope: A slope in which two variables, i.e., variable at x-axis and variable at the y-axis, shows a positive relationship is known as positive slope. In simpler words, a positive slope is one in which the variable x increases with the increase in variable y and/or variable y increases with the increase in variable x.

Negative Slope: A slope in which two variables, i.e., variable at x-axis and variable at the y-axis, shows a negative relationship is known as negative slope. In other words, a negative slope is one in which the variable x increases with the decrease in variable y and/or variable y increases with the decrease in variable x.

A number of absolute values that represent whether a line is steeper or flatter and the direction of the line on the graph are known as a slope or gradient. The slope of a line is fundamental concept in economics and mathematics. It is generally denoted by the letter 'm'. The slope can be calculated by dividing the 'vertical change' with the 'horizontal change' between two distinct points on a line.

There are two main types of slopes which are given below:

Positive Slope: A slope in which two variables, i.e., variable at x-axis and variable at the y-axis, shows a positive relationship is known as positive slope. In simpler words, a positive slope is one in which the variable x increases with the increase in variable y and/or variable y increases with the increase in variable x. Similarly, the variable x decreases with the decrease in variable y, and/or variable y decreases with the decrease in variable x. It means both the variables are complements to each other. A positive slope moves in the upward direction or is upward sloping.

In graphical terms, a positive slope is one in which the line on the graph rises when it moves from left to right. The concept of positive slope can be clearly understood with the help of the supply curve of a producer or firm in economics. The two variables of the curve are price at the y-axis and quantity of goods at the x-axis. Let us assume the firm is producing the goods for profit maximization. Therefore, when the prices of the goods increase, the quantity supplied by the firm of those goods will also increase, while when the prices decrease, the quantity supplied by the firm will decrease.

Negative Slope: A slope in which two variables, i.e., variable at x-axis and variable at the y-axis, shows a negative relationship is known as negative slope. In other words, a negative slope is one in which the variable x increases with the decrease in variable y and/or variable y increases with the decrease in variable x. In the same manner, the variable x decreases with the increase in variable y, and/or variable y decreases with the increase in variable x. This represents an inverse relationship between these two variables. A negative slope moves in the downward direction or is downward sloping.

Graphically, a negative slope is one in which the line on the graph falls when it moves from left to right. One of the best examples of the negative slope of the graph is the demand curve in economics. The two variables of the curve are price at the y-axis and quantity of goods at the x-axis. As we know, the consumers buy a large quantity of a good at a lower price than at a higher price. Therefore, the quantity demanded by the consumers of goods will decrease with an increase in the prices of those goods. On the other hand, when prices of the goods will decrease, the quantity demand will increase.

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The difference between the number of away fans at the semi-final and at the final exists 50.

When b does not equal 0, an ordered pair of numbers a and b, represented as a / b, is said to be a ratio. Two ratios are set to be equal in an equation called a proportion.

To determine how much larger one number is than the other, two numbers of the same unit are compared to determine their connection, which is known as a ratio.

When two quantities of the same unit are compared, the ratio is used to determine how much larger one quantity is than the other. If b is not equal to zero, it is expressed as a/b or a: b.

The number of away fans at the semifinal is

250 × 4/(6 + 4) = 250 × 4/10 = 100

(total number × ratio of a nay fans)

the number of away fans at the final is

500 × 3/(7 + 3) = 500 × 3/15 = 150

(total number × ratio of away fans)

The difference is 150 - 100 = 50

Therefore, the difference between the number of away fans at the semi-final and at the final exists 50.

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Answer:

1/ 3

2/ 1 1/2

3/ -3

4/ 1 1/2

5/ 12

6/ 8

Step-by-step explanation:

1/2(a - 6) + a = 9

1/2a - 3 + a = 9

1 1/2a - 3 = 9

1 1/2a = 12

a= 8

The triangle transitive congruent is proved by considering that if one triangle is congruent to second triangle and second triangle is congruent to third one, then the first triangle is congruent to third triangle.

The transitive property states that if one geometry is congruent with another, and the second is congruent with a third, then the first is also congruent with the third.

Statement : If triangle ABC is similar to triangle PQR and triangle PQR is similar to triangle EFG , then triangle ABC is similar to triangle EFG.

Given, ∆ABC ≅ ∆PQR and ∆PQR ≅ ∆EFG

RTP -> ∆ABC ≅ ∆EFG.

Proof : ∆ABC ≅ ∆PQR, SSA Congruence which implies, AB = PQ --(1)

PR = AC --(2)

and angle ABC = angle PQR --(3)

∆PQR ≅ ∆EFG, AAS Congruence which gives

Angle PQR = Angle EFG --(4)

PQ = EF --(5)

Angle PRQ = Angle EGF --(6)

from (3) and (4) we get,

angle ABC = Angle EFG --(7)

from (1) and (5) we get , AB = EF --(8)

angle BAC = angle FEG = 40° --(9)

Now, from (7) , (8), (9) we conclude that the triangle ABC is Congruent to triangle EFG by ASA Congrauance postulate i.e ∆ABC ≅ ∆EFG.

Hence proved.

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