The table represents a quadratic function. Write an equation of the function in
standard form.

Answer:

19.25 feet

Step-by-step explanation:

Let x be the width of the rectangle in feet. Then, the length of the rectangle is x + 3.5 feet. Since the area of a rectangle is equal to its length times its width, we know that the area of the rectangle is (x + 3.5) * x = 65 square feet.

We can solve for x by setting up the equation (x + 3.5) * x = 65 and then solving for x. This gives us the following:

x^2 + 3.5x - 65 = 0

To solve this quadratic equation, we can use the quadratic formula:

x = (-b +/- sqrt(b^2 - 4ac)) / 2a

In this case, a = 1, b = 3.5, and c = -65, so we have:

x = (-3.5 +/- sqrt(3.5^2 - 4(1)(-65))) / 2(1)

Solving for x, we get:

x = (-3.5 +/- sqrt(12.25 + 260)) / 2

Therefore, the width of the rectangle is x = (-3.5 +/- 19.25) / 2 = -0.75 or 15.75 feet. Since the length of the rectangle is x + 3.5 feet, this means that the length of the rectangle is (-0.75 + 3.5) feet or (15.75 + 3.5) feet, which is equal to 2.75 feet or 19.25 feet.

However, we know that the length of the rectangle must exceed its width by 3.5 feet, so the only solution that is physically possible is x = 15.75 feet and the length of the rectangle is 19.25 feet. Therefore, the length of the rectangle is 19.25 feet.

The leading coefficient is the number that appears before the variable with the highest exponent in a polynomial. It can be either a positive or a negative number.

The leading coefficient is the coefficient, or number, that appears before the variable with the highest exponent in a polynomial. This coefficient helps determine the end behavior of a graph, which means it can either be a positive or a negative number. For example, if the leading coefficient is a positive number, then the graph will have an upward trend on either side of the y-axis, or the x-axis. If the leading coefficient is a negative number, then the graph will have a downward trend on either side of the y-axis, or the x-axis. Additionally, the leading coefficient can also help determine the degree of a polynomial. For example, the leading coefficient of a polynomial with a degree of two will always be a positive number, while the leading coefficient of a polynomial with a degree of three will always be a negative number. In conclusion, the leading coefficient can be either a positive or a negative number and plays an important role in determining the end behavior of a graph and the degree of a polynomial.

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Suppose we're asked to solve the first order linear differnetial equation,

[tex]\longrightarrow\rm{\dfrac{dy}{dx}+y\cdot P(x)=Q(x)}[/tex]

To solve this equation we have to find out a term called 'Integrating Factor' given by,

[tex]\rm{I_f=e^{\int P(x)\ dx}}[/tex]

such that the solution of the equation is,

[tex]\displaystyle\longrightarrow\rm{y\cdot I_f=\int Q(x)\cdot I_f\ dx}[/tex]

Here the given differential equation is,

[tex]\longrightarrow\rm{\dfrac{dy}{dx}-\dfrac{y}{x}=3x}[/tex]

[tex]\longrightarrow\rm{\dfrac{dy}{dx}+y\left(-\dfrac{1}{x}\right)=(3x)}[/tex]

Here,

[tex]\longrightarrow\rm{P(x)=-\dfrac{1}{x}}[/tex]

Integrating wrt x,

[tex]\displaystyle\longrightarrow\rm{\int P(x)\ dx=-\int\dfrac{1}{x}\ dx}[/tex]

[tex]\displaystyle\longrightarrow\rm{\int P(x)\ dx=-\ln|x|}[/tex]

[We will consider constant of integration as zero.]

Then the integrating factor is,

[tex]\longrightarrow\rm{I_f=e^{\int P(x)\ dx}}[/tex]

[tex]\longrightarrow\rm{I_f=e^{-\ln|x|}}[/tex]

Since [tex]\rm{e^{b\,\ln(a)}=a^b,}[/tex]

[tex]\longrightarrow\rm{I_f=x^{-1}}[/tex]

[tex]\longrightarrow\rm{\underline{\underline{I_f=\dfrac{1}{x}}}}[/tex]

Hence the integrating factor for the given differential equation is 1/x.

The number of terms in the expression of this problem is unique.

The expression has only one term.

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.

If we write the expression given below, we can notice that the number of terms is one, and despite having a coefficient and two variables, there are no operational values that separate terms of the expression.

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Answer: D.

Step-by-step explanation:

You can start out with the form AX = B and solve for matrix X that would yield the answer

Then X = (A^-1)(B)

A = [1    -1]

      [1     1]

A^-1 = (1/(1 - (1)(-1)))*[1    1]

                             [-1   1]

which can be written as (1/2) * [1    1]

                                                  [-1   1]

B = [26]

     [6]

(A^-1)(B) = (1/2)*[1    1] [6]

                        [-1   1] [26]

= (1/2)*[32]

          [20]

= [16]

  [10]

The nature of the roots of the quadratic equation 3x² -4x- 2 has  two distinct real roots.

A quadratic equation is a second-order polynomial equation in a single variable x , ax2+bx+c=0. with a ≠ 0 .

The given quadratic equation is 3x² -4x- 2.

Three times of x square minus four times of x minus two.

Nature of Roots depending upon Discriminant.

If the discriminant is equal to zero (b² – 4ac = 0), a, b, c are real numbers, a≠0, then the roots of the quadratic equation ax² + bx + c = 0, are real and equal.

b=-4, c=-2, a=3

b² – 4ac=16-4(3)(-2)

=16+24=40

b² – 4ac>0

The quadratic function has two distinct real roots.

Hence,  the nature of the roots of the quadratic equation 3x² -4x- 2 has  two distinct real roots.

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The spider always moves towards the fly by one unit and the fly moves towards the spider with probability 0.3, away from the spider with probability 0.3, and stays in place with probability 0.4.

(a)

Let x represent the relative location of the spider and fly, where x = 0 when they are in the same position, x = -1 when the spider is one unit away from the fly, x = -2 when the spider is two units away from the fly, and so on. The Markov chain is then given by the following transition probability matrix:

P =

[0.4 0.3 0.3 0 0 0

0.4 0.4 0.2 0.4 0 0

0.4 0.3 0.3 0.4 0.3 0

0 0.4 0.2 0.4 0.3 0.3

0 0 0.4 0.2 0.3 0.3

0 0 0 0.4 0.3 0.3]

(b) The recurrent states of the Markov chain are x = 0 (when the spider and fly are in the same position), x = -1, and x = -2. The transient states of the chain are x = -3, x = -4, and x = -5.

The Markov chain describes the relative location of a spider and fly moving along a straight line in unit increments. The spider always moves towards the fly by one unit and the fly moves towards the spider with probability 0.3, away from the spider with probability 0.3, and stays in place with probability 0.4. The recurrent states of the Markov chain are when the spider and fly are in the same position (x=0), when the spider is one unit away from the fly (x=-1), and when the spider is two units away from the fly (x=-2). The transient states of the chain are when the spider is three units away from the fly (x=-3), four units away (x=-4), and five units away (x=-5).

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7x2 −5x 2 −40x−30

(7x 3−5x+42x−30)+(7x−5)

Add the terms

7x 3+37x−30+(7x−5)

Subtract the terms

7x 3+37x−30+7x−5

Collect like terms

7x 3 +44x−30−5

Solution

7x 3+44x−35

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We can write 'less than 70' as < 70, using the less-than sign.

The less-than sign is a mathematical symbol that denotes an inequality between two values. The widely adopted form of two equal-length strokes connecting in an acute angle at the left, <, has been found in documents dated as far back as the 1560s. In mathematical writing, the less-than sign is typically placed between two values being compared and signifies that the first number is less than the second number. Examples of typical usage include 1⁄2 < 1 and −2 < 0.

Since the development of computer programming languages, the less-than sign and the greater-than sign have been repurposed for a range of uses and operations.

We can use this mathematical symbol to write "less than 70" in a mathematical form.

< 70, indicating that the number is less than 70.

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By using mean formula, the mean number of letters given to each house is 2.1.

The mean, also known as the average, is a statistical measure that represents the central tendency of a set of data. It is calculated by adding up all the values in the data set and dividing by the number of values.

Central tendency is a statistical measure that describes the center of a data set. It is a way to summarize the data and to describe the typical or most common values in the data set.

1x14+2x17+3x0+4x9=84

14+17+9=40

mean = 84/40 = 2.1

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Answer: 40 glasses of syrup

Step-by-step explanation:

Mode is the most common value in the dataset.

The exact values of the trigonometric functions are:

tan (7π / 6) = √3 / 3

cot (7π / 6) = √3

sec (7π / 6) = - 2√3 / 3

csc (7π / 6) = - 2

The angle given in the statement can be represented in a goniometric circle by a ordered pair of the form (x, y) = (cos θ, sin θ) and the expressions of the derivated trigonometric functions can be found by means of the following definitions:

Tangent

tan θ = sin θ / cos θ

Cotangent

cot θ = cos θ / sin θ

Secant

sec θ = 1 / cos θ

Cosecant

csc θ = 1 / sin θ

First, find the exact values of the sine and cosine of the angle:

sin (7π / 6) = sin (π + π / 6) = sin π · cos (π / 6) + cos π · sin (π / 6)

sin (7π / 6) = - sin (π / 6)

sin (7π / 6) = - 1 / 2

cos (7π / 6) = cos (π + π / 6) = cos π · cos (π / 6) + sin π · sin (π / 6)

cos (7π / 6) = - cos (π / 6)

cos (7π / 6) = - √3 / 2

Second, calculate the exact values of the derivate trigonometric functions:

tan (7π / 6) = (- 1 / 2) / (- √3 / 2)

tan (7π / 6) = √3 / 3

cot (7π / 6) = √3

sec (7π / 6) = 1 / (- √3 / 2)

sec (7π / 6) = - 2√3 / 3

csc (7π / 6) = 1 / (- 1 / 2)

csc (7π / 6) = - 2

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The slope-intercept form of the line through (-3,1) parallel to y=-4/3x-1 is y = (-1/2)x - 2 while through (-3,-5) perpendicular to x+2y=-4 is y = 2x + 1.

A straight line on the coordinate plane is represented by a linear function.

A linear function always has the same and constant slope.

The formula for a linear function is f(x) = ax + b, where a and b are real values.

As per the given,

The slope of y = -4/3 x - 1 is -4/3.

Since parallel lines have the same slope thus,

y = (-4/3)x + c

Put (-3,1) into the above equation,

1 = (-4/3)(-3) + c

c = -3

Thus, equation will be y = (-4/3)x - 3

As per the given perpendicular line,

x+2y=-4

y = (-1/2)x - 2

Slope = 2

Put (-3,-5) into the above equation,

-5 = 2(-3) + c

c = 1

Thus, the equation will be y = 2x + 1

Hence "The line through (-3,1) parallel to y=-4/3x-1 has a slope-intercept form of y = (-1/2)x - 2, while the line through (-3,-5) perpendicular to x+2y=-4 has a slope-intercept form of y = 2x + 1".

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The Median of the following observation 65,75 and 80 is 75.

Median is one indicator of central tendency. The value that falls in the middle of the given range of integers is known as the median. In a given data sample, it divides the higher half from the lower half. A minimum of half of the observations fall below or are equal to the median, and a minimum of half of the observations fall above or are equal to the median.

If a data collection has an odd number of observations, the median can be determined as follows:

Step 1: sort the information into ascending or descending order.

Step 2: The middle observation is the median of the given data if the number of observations (let's say n) is odd.

Median = [tex](\frac{n+1}{2} )^{th}[/tex] Observation.

Here n = 3, (3+1)/2 = 2 nd term is the median.

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Expressions that use the inequality symbols to compare two linear expressions are known as linear inequalities.

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Answer:

.25 hours

Step-by-step explanation:

if its $16 to rent a bike for 4 hours, each hour costs 4 dollars meaning that one dollar would be .25 of an hour

Answer: 15 minutes

Step-by-step explanation:

First, take 16 divided by 4 = $4 per hour

We know that 1 hour = 60 minutes

Take 60 divided by 4 = 15 minutes per dollar

15 minutes = 0.25 hour

The common factor of equations 2x , 3x² and 4 is 1

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A , B and C

The value of A = 2x

The value of B = 3x²

The value of C = 4

Now , on simplifying the equations , we get

A = 1 x ( 2 × x )

B = 1 x ( 3 × x × x )

C = 1 x ( 4 )

So , the common term in the equations A , B and C is 1

Therefore , the common factor is 1

Hence , the common factor of the equations is 1

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An equation that depicts a straight line on a graph is said to be linear line.

Linear refers to a straight line. Straight-line graphs called linear graphs are used to show the relationship between two values. The single straight lines on this graph assist to represent a result. The phrase "linear" refers to a straight line and excludes the usage of curves, dots, bars, etc.

A linear line can include more than one variable. Linear lines with two variables, for example, are those that have two variables in them. 2x - 3 = 0 and 2y = 8 are few examples of linear lines.

Steps to draw a linear graph are described below:

Examine the line to see if it is straight or curved. A straight line indicates a linear equation. A nonlinear equation is one that is curved.

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Answer:

Monomial, Binomial,Trinomial,Quadratic equation,Linear equations

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

Credits increase the value of liability, equity, revenue and gain accounts.

Answer:

A. Decrease in assets

Step-by-step explanation:

In this case, the business sold the truck, which represents a decrease in the business's assets. Therefore, the Truck account is credited and the Cash account is debited.

The equations cannot be solved since they are parallel to one another. The equations are hence incoherent.

The supplied equations are 2x3y=7, 4x6y=20, 4x6y=14, and so on. Therefore, the first equation and the second equation are in conflict. As a result, these equations will not have a solution since no value for x or y will satisfy both equations.

The junction of two lines at the spots where a linear equation has a solution are known as the points of intersection. The collection of all feasible values for the variables that satisfy the given linear equation is, in other words, the solution set of the system of linear equations.

There is no solution to the system of linear equations if the graphs of the equations are parallel. No point can be found in this situation when no lines cross one another.

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Answer:

Step-by-step explanation:

To find the roots of the equation x^3 + 2x = x^2 + 2, we need to solve the equation for x. To do this, we can start by subtracting x^2 and 2 from both sides:

x^3 + 2x - x^2 - 2 = 0

This simplifies to x^3 - x^2 + 2x - 2 = 0. Next, we can factor the left side of the equation as follows:

(x - 2)(x^2 + 2x + 1) = 0

Since the product of two factors is equal to 0, this means that at least one of the factors must be equal to 0. Therefore, the roots of the equation are the solutions to the equations x - 2 = 0 and x^2 + 2x + 1 = 0. The first equation has a single solution at x = 2, and the second equation is a quadratic equation that can be solved using the quadratic formula:

x = (-2 +/- sqrt(4 - 411)) / (2*1)

= (-2 +/- sqrt(4 - 4)) / 2

= (-2 +/- sqrt(0)) / 2

= (-2 +/- 0) / 2

= -1 or -1

Therefore, the roots of the equation x^3 + 2x = x^2 + 2 are x = 2, x = -1, and x = -1.

Step-by-step explanation:

the answer to your problem is m= -5/2

Answer:

-2.5

Step-by-step explanation:

The slope formula [tex]\frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex] tells us the formula for any slope with 2 given points. We have to plug in the numbers from our 2 ordered pairs to get [tex]\frac{-4 - 1}{2 - 0}[/tex], which simplifies to [tex]\frac{-5}{2}[/tex] and can be written as -2.5

The slope of this equation is -2.5

Answer: 19.625 ft^2

Step-by-step explanation: To calculate the area of a circle you have to use Area=pi*r^2, so the radius is 2.5 ft, Area = 3.14*2.5^2 = 19.625 ft^2

Answer:

Step-by-step explanation:

Done the I-ready.

A dataset's mean (also known as the arithmetic mean, which differs from the geometric mean) is calculated by dividing the sum of all values by the total number of values. The term "average" is frequently used to describe this measure of central tendency.

What does the mode mean?

By summing all the numbers in a data collection and dividing by the total number of values in the set, the mean (average) of the data set can be determined.

              A data collection is ordered from least to largest, and the median is the midpoint number. A data set's mode is the number that appears the most frequently.

What are the mean, median, and example?

The most frequent number, or the one that happens the most frequently, is known as the mode.

                      Example: Since the number 2 appears three times, more than any other number, it is the mode of the numbers 4, 2, 4, 3, and 2.

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The end behavior of a function f describes the behavior of the graph of the function at the "ends" of the x-axis. In other words, the end behavior of a function describes the trend of the graph

What is the End Behavior of a Function?

When a function f (x) increases or decreases without bound, the behaviour at its finish is referred to as the function's behaviour.  increases or decreases without bound. In other words, the end behavior describes the ultimate trend in the graph of f(x) as we move towards the far right or far left of the x-axis.

End behavior is represented in mathematical notation by symbols that indicate the impact on the function when the variable goes toward plus or negative infinity. For instance, the claim

The expression f(x)+ as x+ indicates that the function will grow unboundedly huge as the variable rises and can be read as "f(x) approaches positive infinity as x approaches positive infinity."

Because it can be considered as defining the long-term behavior of a quantity of interest, the concept of the end behavior of a function has practical applications. For instance, epidemiologists might be interested in predicting future trends in the prevalence of a disease. Whether or not case numbers are expected to plateau or keep rising is a sign of whether or not the epidemic is currently under control.

so ,The end behavior of a function f describes the behavior of the graph of the function at the "ends" of the x-axis.

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The behavior of the function graph at the "ends" of the x-axis is known as the end behavior of a function, or f. In other words, the graph's trend is described by the function's final behavior.

The formula f(x)+ as x+, which may be translated as "f(x) approaches positive infinity as x approaches positive infinity," denotes that the function will become unboundedly massive as the variable increases.

so ,

The behavior of the function graph at the "ends" of the x-axis is known as the end behavior of a function, or f.

Answer:

5

Step-by-step explanation:

280 divided by 56 = 5

to check your answer just 5x56=280

hope this helped

As per the given suspended mass, the function that makes values of both x and θ develop the maximum possible moment at A is written as  14715 Nm (clockwise)

While looking into the given question, we have identified the following values, adjusted for any angle 00 ≤ θ ≤ 900 any extension 0 ≤ x ≤ 5 m.

Now, Let us consider an equation to calculate the moment using θ and x as variables around point A.

=> ((9−1.5) + x) (cosθ) (120) (9.81)

When we simplifying the given equation gives us:

=> MA ​= (7.5+x) 1177.2 cosθ

 Here the maximum moment will develop if θ = 0 and x=5 m.

Now, by substituting these values gives us:

=> MA​=(7.5+x)1177.2cosθ

=> MA​=(7.5+5)1177.2cos00

Therefore, the value of MA is written as, 14715 Nm (clockwise)

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Standard deviation (S.D) and Mean are 14.603 and 19.5 respectively.

Given data

16, 24, 18, 17, 12, 17, 25, 15, 19, 20, 27, 24

here we need to find standard deviation of the given data from mean

standard deviation of a data = ∑x²/ N

here ∑x² = sum of squares of deviations of the data

         N = number of observations    

⇒ Mean of the data = [16+24+18+17+12+17+25+15+19+20+27+24]/ 12  

                                =  234 / 12 = 19.5  

⇒ now calculate deviations in following table

    observations          deviations                 squares of deviations

        16                              3.5                                  12.25

        24                             4.5                                  20.25

        18                               1.5                                    2.25

        17                               2.5                                   6.25

        12                               7.5                                  56.25

        17                                2.5                                  6.25

        25                               5.5                                  30.25

        15                               4.5                                  20.25

        19                               0.5                                   0.25

        20                              0.5                                   0.25

        27                               0.7                                   0.49

        24                               4.5                                 20.25

      N = 12                                                            ∑ x² = 175.24  

 standard deviation (S.D)  = 175.24/ 12 = 14.603    

Hence Standard deviation (S.D) and Mean are 14.603 and 19.5 respectively.

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Answer:

x = 60

Step-by-step explanation:

We are told that the figures are similar.  That means the that corresponding sides are proportional.

Sides Ps and ZW are corresponding sides.  We can use them to find the scale factor form the smaller figure to the larger figure.

40 times some number (the scale factor) is 75

Let x = the scale factor

40x = 75  Divide both sides by 40

[tex]\frac{40x}{40}[/tex] = [tex]\frac{75}{40}[/tex]

x = 1.875  This is are scale factor.

Take side YZ and multiply its length of 32 by the scale factor to find x.

32(1.875) = 60

Answer:

x = 60

Step-by-step explanation:

assuming you mean PQRS is similar to WXYZ , then ratios of corresponding sides are in proportion, that is

[tex]\frac{RS}{YZ}[/tex] = [tex]\frac{PS}{WZ}[/tex] ( substitute values )

[tex]\frac{x}{32}[/tex] = [tex]\frac{75}{40}[/tex] ( cross- multiply )

40x = 2400 ( divide both sides by 40 )

x = 60