What is the quotient and remainder, written as partial fractions, of
15x²+52x +43
3x²+5x-8

By making the discriminant equal to zero, we can see that the quadratic equation has one real solution only for z = 10.

For a general quadratic equation:

a*x^2 + b*x + c = 0

We define the discriminant as:

D = b^2 - 4ac

The quadratic equation will have only one real solution only if the discriminant is equal to zero.

Here the quadratic is:

(z - 5)*x^2 - z*x + 5 = 0

Then the discriminant is:

(-z)^2 - 4*5*(z - 5)

And that must be equal to zero, then we need to solve:

z^2 - 20*(z - 5) = 0

z^2 - 20z + 100 = 0

Using the quadratic formula we will get:

[tex]z = \frac{20 \pm \sqrt{(-20)^2 - 4*1*100} }{2} \\\\z = \frac{20 \pm 0}{2} \\\\z = 10[/tex]

So the quadratic equation has one real solution only for z = 10.

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The fraction that correctly represents the ratio of 5 to 15 in it's simplest form is option C. 1/3.

Fractions are type of numbers which are written in the form p/q, which implies that p parts in a whole of q.

Here p, called the numerator and q, called the denominator, are real numbers.

The given ratio is 5 to 15.

Fraction in simplest form means that, the numerator and denominator cannot be factored by a common factor again.

In order to find the given ratio in simplest form, find the greatest common factor of both 5 and 15.

5 = 5

15 = 3 × 5

5 is the only common factor and so the greatest common factor of both 5 and 15.

5 / 15 = (5 × 1) / (5 × 3)

         = 1/3

Hence 1/3 is the simplest form of the ratio 5 to 15.

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Your question is incomplete. The correct question is as follows.

Chose the fraction that correctly represents the ratio 5 to 15 in simplest form.

A. 2/3

B. 10/30

C. 1/3

D. 1/5

He then constructs a graph to visualize his results,  For a given number of selections, we can use Joseph's graph to find the number of diamonds that he has picked so far is FALSE.

Probability refers to potential. A arbitrary event's circumstance is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to read the liability of colorful events. The degree to which commodity is likely to be is principally what probability means.

You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.

We are only interested in finding red card so we can't differentiate between diamond or hearts at any time. Hence,

The false statement is:

For a given number of selections, we can use Josephs graph to find the number of diamonds that he has picked so far.

Option B is correct.

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Answer:

  y = 1/8x +1/64

Step-by-step explanation:

You want the equation of the line that is tangent to the curve y = x³ -x⁴ in exactly two places.

Since the tangent line is not crossed, the difference between the line and the given curve is a 4th-degree polynomial with two roots, each of multiplicity 2. If those are 'a' and 'b', we can write for line y = mx +p, ...

  mx +p -(x³ -x⁴) = (x -a)²(x -b)² = 0

  x⁴ -x³ +0x² +mx +p = x⁴ -2(a+b)x³ +(a²+4ab+b²)x² -2ab(a+b)x +a²b²

Equating coefficients, we have ...

The coefficient of x³ tells us ...

  a +b = 1/2

The coefficient of x² tells us ...

  0 = (a +b)² +2ab

  0 = (1/2)² +2ab

  ab = -1/8

Using these value, we can find the equation of the line to be ...

  m = -2ab(a+b) = -2(-1/8)(1/2) = 1/8

  p = (ab)² = (-1/8)² = 1/64

The equation of the line is ...

  y = 1/8x +1/64

__

Additional comment

To find the line, we did not need the coordinates of the points of tangency. They are ((1-√3)/4, (3-2√3)/64) and ((1+√3)/4, (3+2√3)/64).

The slope of the line is equal to the slope of the quartic at the point of inflection of its derivative, at x=1/4.

Answer:

Step-by-step explanation:

Let's call the number of friends "f". Since each friend paid $12.75 for their meal, the cost of the meals can be represented as f * 12.75. And since each friend left the same amount for a tip, the cost of the tips can be represented as t * f.

The total cost of the dinner including the tip was $61, so we can write the following equation:

f * 12.75 + t * f = 61

Expanding the left side of the equation:

12.75f + t * f = 61

Combining like terms:

13.75f + t = 61

Solving for t:

t = 61 - 13.75f

So, the equation that describes the situation is t = 61 - 13.75f.

Answer:

Y> (x - 4)² + 28

Step-by-step explanation:

The total cost of the ingredient to make the cake and the comparison to the percentage discount to get the best deal indicates that we get;

A percentage is the representation of a proportion of an item per 100 of a set of items.

Quantity of flour = 5 lb

Unit cost for the flour = $2.74 per lb

Quantity of sugar = 2 1/2 lb

Unit cost of the sugar = $1.70 per lb

Quantity of butter = 2 lb

Unit cost of the butter = $7.49 per lb

Quantity of eggs required = 1/2 dozen

Cost of six eggs = $4.85

Miss Brown's total bill for the four items, C, is therefore;

C = 5 × 2.74 + (2 1/2) × 1.7 + 2 × 7.49 + 4.85 = 37.78

The discount available for purchases of $50 or more = 5% discount on the purchase

The amount of more money she will have to spend to get the 5% discount is therefore;

$50 - $37.78 = $12.22

Miss Brown will have to spend $12.22 extra to get the 5% discount

The 5% discount indicates that spending $50, Miss Brown only has to pay $50 - $50 × 5% = $47.5

Therefore, in order to get the discount, the total amount spent on the purchase is increase by $47.5 - $37.78 = $9.72, and the amount saved from the discount is $2.5, therefore;

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On solving the provided question, we can say that the trigonometry sin3θcos2θ+cos3θsin2θ = 4sin3(1−2cos2θ)+sinθ

The area of mathematics knοwn as trigonometry examines the correlation between triangle side lengths and angles. The area first appeared in the Hellenistic era, arοund the third century BC. frοm the use of geometry in astronοmical study.

The area of mathematics knοwn as exact methods deals with specific trigonometric functions and hοw they might be used in calculations. There are six popular trigonοmetric functions in trigonometry. Sine, cosine, tangent, cοtangent, secant, and cosecant are their respective names and acronyms (csc). Studying the characteristics οf triangles, particularly right triangles, is called trigonοmetry. The study of geοmetry, however, is the characteristics of all geοmetric figures.

the trigonometry

sin3θcos2θ+cοs3θsin2θ=4sin3(1−2cos2θ)+sinθ

Use the variation cοs2θ=1−sin2θ to express sin(5θ) in terms of sinθ

sin5θ=4sin3θ(1−2(1−sin2θ))+sinθ(8(1−sin2θ}

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For the function y = f(x). The statements are -

A. False

B. True

C. False

D. False

What is a function?

In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.

The data points for the function y = f(x) is given.

The domain of a function is the set of "valid inputs" to the function, while the range of a function is the set of "all outputs" of the function.

The function y = f(x) represents the relation between the set of elements in the domain of f and the set of elements in the range of f, where the variable x represents an element from the domain of the function and f(x) represents an element of the range of the function.

The first row in the table lists all the inputs to the function f. The second row lists the outputs of f corresponding to the inputs in the first row.

(a) The given statement is false because the range of f is the set of all f(x) where x is in the domain of f: {-5,-3,0,2,6,7,9,10,13}

(b) The given statement is true because, according to the table, the given input x = -3 has an output of f(x)=2; that is, f(-3) = 2.

(c) The given statement is false because the domain of f is the set of all values of x for which f(x) is defined:

{1,2,30,1,2,3,0,1}

(d) The given statement is false because, according to the table, the given input x = 0 has an output of f(x)=3; that is, f(0) = 3.

Therefore, the three statements are false and one is true.

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Answer:

75

Step-by-step explanation:

Use the arithmetic progression formula:  S =n/2(2a + (n-1)d)

a = 5, d = 5, n = 5

S = 5/2((2(5) + (5-1)(5)) = 75

The quadratic function is of the given question is f(x) = [tex]-2x^{2}[/tex] + 10x + 1.

When a, b, and c are constants and x is an unknown variable, the equation has the form [tex]ax^{2}[/tex] + bx + c = 0, which is known as a quadratic equation. The unknown variable x is solved for using the quadratic equation.

The largest power of the variable in a quadratic equation is two. The variable is now squared, but it may also have a coefficient, often known as a multiplier, in front of it. Two more components in the equation are made up of a constant and a variable with an exponent of 1.

In the given question,

The equation of the quadratic function with a vertex at (-1, 3) and passing through (-4, -6) can be written as:

y = [tex]a(x + 1)^{2}[/tex] + 3

Where a is a constant that determines the shape of the function. To find the value of a we can substitute the coordinates of the vertex and the point into the equation and solve for a.

Substituting (x, y) = (-1, 3) into the equation:

3 = [tex]a(-1 + 1)^{2}[/tex] + 3

Simplifying:

3 = [tex]a(0)^{2}[/tex] + 3

3 = 3

Since both sides are equal, a = 1.

Substituting (x, y) = (-4, -6) into the equation:

-6 = [tex]a(-4 + 1)^{2}[/tex] + 3

Simplifying:

-6 = [tex]a(-3)^{2}[/tex] + 3

-6 = 9a + 3

-9 = 9a

Divide each side by 9:

-1 = a

Since both values of a are equal, a = -1.

Therefore, the equation of the quadratic function with a vertex at (-1, 3) and passing through (-4, -6) is:

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

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The evaluation of the function expressions gives:

(q + f)(5) = 1

(f - q)(2) = -2

(q f)(4) = 9

(f /q)(0) = 1

A function is an expression that shows the relationship between the independent variable and the dependent variable.  A function is usually denoted by letters such as f, g, etc.

From the graph, when c = 5, q(5) = 0, f(5) = 1. Thus:

(q + f)(5) = 0 + 1 = 1

From the graph, when c = 2, f(2) = 0, q(2) = 2. Thus:

(f - q)(2) = 0 - 2 = -2

From the graph, when c = 4, q(4) = 3, f(4) = 3. Thus:

(q f)(4) = 3 * 3 = 9

From the graph, when c = 0, f(0) = 4, q(0) = 4. Thus:

(f /q)(0) = 4/4 = 1

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Answer: The outlier is $95 and the plot is D.

Step-by-step explanation:

          An outlier is the data value that's much larger or much smaller than the other data values. Here, that is $95.

          Excluding this value, our smallest is $15 and the largest is $60. We plot a dot at both of these points.

          The middle dot is the median. To find this, we will arrange the values in order and find the middle value. (We are excluding $95.) After doing this process, we see the middle dot is at 39.

    $15, $20, $38, $60, $45, $40, $35, $50

    $15, $20, $35, $38, $40, $45, $50, $60

    ----, ----, ----, $38, $40, ----, ----, ----

                    [tex]38 + 40 = 78, \;\;78 / 2\;\;\boxed{= 39}[/tex]

          This means we have dots at 15, 39 and 60, leading us to answer D.

On solving the provided question, we cans ay that the equation is 4(8x + 2) = 4(7x + 2 + x); when we put x = , LHS = RHS

A mathematical equation is a formula that joins two statements and uses the equal symbol (=) to indicate equality. A mathematical statement that establishes the equality of two mathematical expressions is known as an equation in algebra. For instance, in the equation 3x + 5 = 14, the equal sign places the variables 3x + 5 and 14 apart. The relationship between the two sentences on either side of a letter is described by a mathematical formula. Often, there is only one variable, which also serves as the symbol. for instance, 2x – 4 = 2.

the equation is

4(8x + 2) = 4(7x + 2 + x)

32x + 8 = 28x + 8 + 8x

36x - 32x = 0

4x = 0

x = 0

when we put x = , LHS = RHS

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The sequence will be aₙ=$1250-118ⁿ⁻¹ where n= place at which a bowler finished.

What exactly is a sequence?

A sequence is a collection of numbers that follows pattern. Olivia, for example, has been offered a position with a beginning monthly pay of $1000 and a yearly raise of $500. To figure out her monthly pay for the first three years? The prices will be $1000, $1500, and $2000. Note that Olivia's income over a number of years creates a sequence as they follow a pattern where the numbers increase by $500 each time.

Now,

given award money for 1, 2, 3 places is given as $1250, $1132 and $1014

that means for the sequence, first term is 1250 and difference between the next terms will be 118 (less)

So, the sequence will be aₙ=$1250-118ⁿ⁻¹

where

n=place at which a bowler is placed.

Hence,

           The sequence will be aₙ=$1250-118ⁿ⁻¹.

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Answer:

541.5 ksh

Step-by-step explanation:

The equation that best represents the curve of best fit for this set of data is y = ae^{bx}, where a and b are constants to be determined by the data points.

An equation is a mathematical expression that uses symbols, such as numbers and variables, to represent a relationship between two or more values or quantities. Equations are used to describe physical laws, solve problems and to make predictions. Equations are fundamental to all areas of mathematics and science, and their use is essential for understanding and exploring the natural world.


This equation is an example of an exponential function, which is used to model data that increase or decrease at an increasing rate. The constants a and b can be determined using a least-squares regression to find the line of best fit. The equation can then be used to extrapolate future values from the data points.

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On solving the provided question, we can say that  inches if a cylinder's enlarged capacity is 45 in3, its compressed volume is 6 in cubic .

One of the most fundamental curved geometric forms is the cylinder, which is often a three-dimensional solid. It is referred to as a prism with a circle as its base in elementary geometry. Several contemporary fields of geometry and topology also define a cylinder as an indefinitely curved surface. A three-dimensional object known as a "cylinder" consists of curving surfaces with circular tops and bottoms. A cylinder is a three-dimensional solid figure that has two bases that are both identical circles joined by a curving surface at the height of the cylinder, which is determined by the distance between the bases from the center. Examples of cylinders are cold beverage cans and toilet paper wicks.

In a certain engine, the compression ratio is 7.5: 1.

A cylinder's enlarged volume is 45 in3 in.

We must determine the cylinder's compressed volume.

Let v represent the cylinder's compressed volume.

The aforementioned circumstance can be described as

7.5 : 1 = expanded volume : compressed volume

7.5/1 = 45/v

7.5 = 45/v

v = 45/7.5

v = 6 cubic inches .

Therefore, if a cylinder's enlarged capacity is 45 in3, its compressed volume is 6 in cubic .

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The value of the length /MK/ is 10 units

Recall that in an isosceles reangle, two sides are equal and the base angles are equal

We can use the cosine rule to find the length MK

N² = K²+M² - 2KMCosN

This implies that /MK/² = 5²+5²-2*5*5*cos110

⇒25+25-50*-0.9990

= 50+49.95

=99.95  this can be approximated to 100

the /MK/ = √100= 10 units

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The measures of the center or variability that are greater than 5 degrees are :

The mean of the data on temperature this week and the last is :

= ( 4 + 10 + 6 + 9 + 6 ) / 5

= 7 degrees Fahrenheit 4+ 10+ 6+9+ 6

The mean of the data on temperature last week and the last is :

= ( 13 + 9 + 5 + 8 + 5 ) / 5

= 8 degrees Fahrenheit

The range of this week's temperature is :

= 10 - 4

= 6 degrees Fahrenheit

All these measures of center are above 6 degrees.

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The full question is:

Temperatures for 5 Days This Week and Last Week

Low Temperatures This Week (Degrees Fahrenheit) 4 10 6 9 6 Low Temperatures Last Week (Degrees Fahrenheit) 13 9 5 8 5 Which measures of center or variability are greater than 5 degrees? Select three choices.

a) the mean of this week’s temperatures

b) the mean of last week’s temperatures

c) the range of this week’s temperatures

d)the mean absolute deviation of this week’s temperatures

e) the mean absolute deviation of last week’s temperatures

Explanation:

In this case the median value is more representative of the 'center' of the population. The median is a better measure when the data set is skewed, has fat tails as compared to a normal didtribution, or has outliers. Otherwise you should use the arithmetic mean.

The graph of the system of the equation y > -1, y≥ x²+ 1, and (x-1)²+(y-1)² ≤ 16 is attached with the feasible region of each part shaded

The equation y > -1 is a horizontal line drawn at point y = -1 and shaded above the line because the sign is greater than. the line is dashed line line since the equation is not "greater than or equal to".

The equation y ≥ x²+ 1 is a parabolic equation

The equality sign attached to the greater than makes the line a solid line and because it is greater than the shading is inside the curve.

The equation (x - 1)² + (y - 1)² ≤ 16 is a equation of a circle

The equality sign attached to the greater than ("greater than or equal to") makes the line a solid line and because it is greater than the shading is inside the circle

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Answer:

After rounding the nearest tenth of a square foot, the surface area of the  circular cylinder is 131.88 feet^2.

Step-by-step explanation:

To find surface area of a right circular cylinder we can use the formula,

A = 2πrh + 2πr^2

Here A is surface area,

π is mathematical constant ,

r is radius, and h is the height of the cylinder.

We have

r = 3 ft

h = 4 ft,

Now we put the values to find the surface area,

A = 2π × 3 × 4 + 2π ×3^2

A = 24π + 18π

A = 42π

A= 42×3.14                             (we know π = 3.14 )

A = 131.88

After rounding the nearest tenth of a square foot, the surface area of the  circular cylinder is 131.88 feet^2.

Answer:

Let's call the number of 7,000 gallon tank cars "x1", the number of 14,000 gallon tank cars "x2", and the number of 28,000 gallon tank cars "x3".

We can write the following constraints based on the total carrying capacity of the fleet:

7,000x1 + 14,000x2 + 28,000x3 = 532,000

x1 + x2 + x3 = 25

x1, x2, x3 >= 0 (since we can't have a negative number of tank cars)

This is a linear programming problem, which can be solved using simplex method or the graphical method.

By using the graphical method, we find that there are multiple optimal solutions. For example, one solution is x1 = 11, x2 = 7, and x3 = 7. This means that the manufacturer should lease 11 7,000 gallon tank cars, 7 14,000 gallon tank cars, and 7 28,000 gallon tank cars to meet the total carrying capacity requirement of 532,000 gallons.

So, the answer to the question is: There are multiple possible combinations of how the tank cars should be leased. The combinations are obtained from the equations x1 = 11 + 7t, x2 = 7 + 7t, and x3 = 7 + 7t.

Answer:

A. (f+g)(x)=-2x^2-2x+8

Step-by-step explanation:

You combine like terms by adding

The graph of y = 2 sin(x) + 2 is given by the following option:

Graph A.

The standard definition of the sine function is given as follows:

y = Asin(B(x - C)) + D.

For which the parameters are given as follows:

The function in this problem has an amplitude of 2, meaning that before the vertical shift, it oscillates between -2 and 2.

Due to the vertical shift, the function is going to oscillate between 0 and 4, with a period of 2π and no phase shift, crossing the midline of y = 2 at the origin, hence Graph A is the correct graph.

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Answer:

To estimate the number of seventh-graders who attend summer camp, we can use the sample proportion of 30 out of 200 students.

Let's call the population proportion of seventh-graders who attend summer camp "p".

The sample proportion can be estimated as:

p = 30/200 = 0.15

To estimate the number of seventh-graders in the entire school who attend summer camp, we can multiply the population size by the sample proportion:

Estimated number of seventh-graders attending summer camp = p * population size = 0.15 * 780 = 117

So, about 117 seventh-graders in the school are expected to attend summer camp.

Answer:

117 seventh-graders in the school are expected to attend summer camp.

Step-by-step explanation:

The system of equation is written as follows

y = 90 + x

y = 40 + 6x

With regards to cost the choices will be equivalent in 10 day

Assuming additional day is x

Felipe's Moving, they would pay $90 for the first day plus $1 per additional day:

= 90 + x

Rent-a-Truck, the family would pay $40 for the first day plus $6 per additional day.

= 40 + 6x

Equating both costs gives

90 + x = 40 + 6x

90 - 40 = 6x - x

50 = 5x

x = 10

in 10 days the cost will be same

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(1) The cost of the phone call was less than $4.16

(2) The cost of the phone call was greater than $3.35

While this is a seemingly simple min/max problem, I was wondering if there is a faster way to deal with decimals—as it took me ~4mins to arrive at the solution.

The cost of a certain phone call was $0.75 for the first 3 minutes and $0.20 for each additional minute after the first 3 minutes. Did the phone call last longer than 15 minutes

According to the above the cost for n minutes, where n>3, is 75 + (n−3)∗20=20n+15

cents. W need to find whether n>15. The cost for 15 minutes is 20n+15=20∗15+15=315

, so we need to find whether the cost is greater than 315 cents.

(1) The cost of the phone call was less than $4.16. Not sufficient.

(2) The cost of the phone call was greater than $3.35. Sufficient.

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To solve the problem we would use tangent as a trigonometric function and find the unknown value.

Simply put, trigonometric functions—also referred to as circular functions—are the functions of a triangle's angle. This means that these trig functions provide the relationship between the angles and sides of a triangle. Sine, cosine, tangent, cotangent, secant, and cosecant are the fundamental trigonometric functions.

From the given triangle we observe that the given values are:

Angle B, side a and side b.

To find any of the unknows in the three we use the trigonometric function tangent.

Tan α = opposite side / adjacent side

For the given figure:

Tan B = b / a

Hence, to solve the problem we would use tangent as a trigonometric function and find the unknown value.

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