X= (3, 3², -2, 6) : F(x)= 2/3 x + 3

We can see (either graphically or algebraically) that the lowest point, or minimum, occurs when x = 3 and y = −4. That is the location of the vertex.

The coordinate of the vertex is (3 , −4)

given equation

y = [tex]x^{2}[/tex] - 6x + 5

  = [tex](x - 3)^{2}[/tex] - 4

We can see (either graphically or algebraically) that the lowest point, or minimum, occurs when x = 3 and y = −4. That is the location of the vertex.

The coordinate of the vertex is (3 , −4)

quadratic function :

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero.

The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape. The picture below shows three graphs, and they are all parabolas.

All parabolas are symmetric with respect to a line called the axis of symmetry. A parabola intersects its axis of symmetry at a point called the vertex of the parabola.

You know that two points determine a line. This means that if you are given any two points in the plane, then there is one and only one line that contains both points. A similar statement can be made about points and quadratic functions.

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The decimal that should be represented as a negative rate in this problem is the 2.2 millimeters per year because it represents a decrease in water level.

The two factors that Ericka should multiply to find the total variation in water level are the rate of change (2.2 millimeters per year) and the time period over which the change occurred (6.2 years). In other words, she should multiply 2.2 by 6.2 to find the total variation in water level.

A novel and somewhat paradoxical tool for monetary policy, negative interest rates are not common. The goal of negative interest rates is to encourage lending and spending rather than saving and hoarding. When cash is placed with a bank, it results in a storage fee rather than the chance to receive interest income. When cash is placed with a bank, it results in a storage fee rather than the chance to receive interest income.

The quantity you are measuring is said to be increasing over time if the rate of change is positive, whereas the opposite is true if the rate of change is negative.

The complete question is:-

Ericka is also in Mr. Whittaker’s science class and she recorded the variation in water level at a different part of the river. She observed that the water level there fell 2.2 millimeters per year over a period of 6.2 years This data is compared to the average annual trend, which shows the water level rising 1.8 mm/year. Which of the decimals in the problem should be represented as a negative rate? Which two factors should Ericka multiply to find the total variation in water level?

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The area of the region bounded by the given curves is   [tex]32lnx-\frac{58}{3}[/tex]

Given curves,

y = [tex]6x^{2} lnx[/tex]

y = 24lnx

We will find the intersection points of these two curves by equating and then solving.

The intersection of these two curves will be at x = 1,2.

The area under the curves,

A = [tex]\int\limits^2_1 {(24lnx-6x^{2} lnx)} \, dx[/tex]

To solve, concentrate on the unbounded (indefinite) integral:

a = [tex]\int\limits {(24lnx-6x^{2} lnx)} \, dx[/tex]

  = [tex]-6\int\limits {(lnx)(x^{2} -4)} \, dx[/tex]

We will use integration by parts to solve this integration.

[tex]\int\limits {u} \, dv=vu-\int\limits{v} \, du[/tex]

Let us assume,

u = lnx and v = [tex]\frac{1}{3}x^{3} -4x[/tex]

Therefore,

[tex]du = \frac{1}{x}dx[/tex]

[tex]dv=x^{2}-4[/tex]

Putting the values,

a = [tex]-6[(lnx)(\frac{1}{3}x^{3}-4x) - \int\limits{(\frac{1}{x} )(\frac{1}{3}x^{3}-4x)} \, dx ][/tex]

  = [tex]-6[(lnx)(\frac{1}{3}x^{3}-4x) - (\frac{1}{9}x^{3}-4x)][/tex]

  = [tex]-6(lnx)(\frac{1}{3}x^{3}-4x) + \frac{2}{3}x^{3}-24x[/tex]

Returning to the definite integral, now

A = [tex]\int\limits^2_1 {(24lnx-6x^{2} lnx)} \, dx[/tex]

   = [tex][-6(lnx)(\frac{1}{3}x^{3}-4x) + \frac{2}{3}x^{3}-24x]^{2} _{1}[/tex]

   = [tex]-6ln2(\frac{8}{3}-8)+\frac{16}{3}-48-\frac{2}{3}+24[/tex]

   =  [tex]32lnx-\frac{58}{3}[/tex]

Hence, The area of the region bounded by the given curves is [tex]32lnx-\frac{58}{3}[/tex].

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If there are no duplicate digits, then and only then, is a number considered to be unique. For instance, unique numbers include 20, 56, 9863, 145, etc., but non-unique numbers include 33, 121, 900, 1010, etc.

Any two items that are the only ones of their kind are equally unique. Unique is defined as the only one like it. There is not a singular most. Each number is distinct because there is only one other one like it. For instance, if you subtract a three-digit number, like 345, from its reverse, 543, you get a difference of 198.

If the number An'obtained is subtracted from a number An made up of n consecutive digits in ascending order by reversing the digits of An, then the difference is always a constant. This constant is termed as the Unique number Un.

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By resolving these equations, we can arrive at one solution.

To find the number of solutions to a system of equations, you can solve the system either by graphing the equations or by using algebraic methods such as substitution or elimination.

If the equations are compatible (meaning that there is at least one combination of values for the variables that satisfies both equations), then the system has at least one solution. If the equations are independent (meaning that they are not related to each other and cannot be derived from each other), then the system has exactly two solutions. If the equations are dependent (meaning that one of the equations can be derived from the other), then the system has exactly one solution. If the equations are incompatible (meaning that there is no combination of values for the variables that can satisfy both equations), then the system has no solutions.

In this case, the equations are 3x + 2y = 5 and 2x - 3y = 7. To find the number of solutions to this system, you can either graph the equations or solve the system using algebraic methods.

Using algebraic methods, you can start by solving one of the equations for one of the variables in terms of the other:

3x + 2y = 5

2y = -3x + 5

y = (-3/2)x + (5/2)

Then, you can substitute this expression for y into the other equation to solve for x:

2x - 3((-3/2)x + (5/2)) = 7

2x + (9/2)x - (15/2) = 7

(11/2)x - (15/2) = 7

(11/2)x = (15/2) + 7

(11/2)x = (15 + 14)/2

(11/2)x = 29/2

x = 29/11

Substituting this value for x back into the first equation, you can solve for y:

3(29/11) + 2y = 5

87/11 + 2y = 5

2y = 5 - 87/11

2y = (55 - 87)/11

2y = -32/11

y = -32/22

Therefore, the solution to this system of equations is x = 29/11 and y = -32/22. Since there is only one solution, this system has exactly one solution.

The complete question is:-

How many solutions can we get by solving these equations 3x + 2y= 5;2x -3y =7?

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Answer:

  a) 81/500

  b) 7/90

  c) 3/11

Step-by-step explanation:

You want to write the fractions that correspond with the decimals ...

  a) 0.162

  b) 0.0777...

  c) 0.272727...

The fraction corresponding to a terminating decimal can be written based on how you pronounce the fraction.

  0.162 = "one hundred sixty-two thousandths" = 162/1000

This can be reduced by removing a common factor of 2.

  = 81/500

The fraction equivalent to a repeating decimal can be found by a multiply and subtract process. In general, this makes the denominator be a number that has as many 9s as the decimal has repeating digits.

This has one repeating digit, 7. This means we want to convert to a fraction by multiplying by 10 to the power of 1:

  x = 0.0777...

  10x = 0.7777...

Now, we can subtract the original number:

  10x -x = 0.7777... -0.0777... = 0.7000...

Notice that the repeating digits cancel. Now we have ...

  9x = 0.7 = 7/10

Dividing by 9 gives ...

  x = 7/90

Using the above method with 10 to the power of 2 (with a 2-digit repeat), we get ...

  100x = 27.272727...

  100x -x = 27.272727... -0.272727... = 27

  99x = 27

  x = 27/99 = 3/11

__

Additional comment

In general, the repeating part of a repeating decimal can be expressed over an equivalent number of 9s. For example, the 6-digit repeat 0.142857142857... is equivalent to 142857/999999. As it happens, this one reduces to 1/7.

If there are intervening digits, as in 0.4333... for example, then they can be treated as a terminating decimal, and the repeating part can be multiplied by an appropriate power of 10 to account for the shift from the decimal point:

  0.4333... = 0.4 + 0.1 × 0.333...

  = 4/10 + 1/10 × 3/9 = 4/10 + 1/30

  = 12/30 +1/30 = 13/30

The rule of the reflection of the triangle ABC to A'B'C' across the y-axis will be (x, y) to (-x, y).

It is the image of the point which is located in the opposite direction of a given point.

The rule of the refection will be given as,

Firstly, the rule for reflecting a point about the y-axis is;

While reflecting on the y-axis, we get the reflected points (x, y) by the coordinates (-x, y).

The standard of the impression of the triangle ABC to A'B'C' will be (x, y) to (- x, y).

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Answer:

Step-by-step explanation:

Domain is constant since there are no value that x could be to make this not a possible answer

First f(x)=3x+5 y=3x+5 then y+5=3x so (y+5)/3

inverse is (x+5)/3

Then g(x)=x^2+6 y=x^2+6 then y-6=x^2 so then x=  \sqrt{y-6}

inverse is   \sqrt{y-6}

No, a square root is not always an irrational number because the perfect square roots are rational number.

Any real number that cannot be written as p/q, where p and q are both integers, the quotient of two integers.

If a number is not a perfect square, the square root of that number is not rational.

For example, the square root is 4 is 2.

The square root of 9 is 3.

But whereas the square root of 5 is 2.23606797755555...... which is irrational

The square root of 8 is 2.8284271247555..... which is irrational.

So, therefore square root of a number is not irrational always it can be rational also.

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Answer:

Below

Step-by-step explanation:

6 (x-1.50) = 51        6 = number of tix     x - 1.50 = sale price     x = reg price

6x - 9 = 51

6x = 60

x = 10 dollar regular price

Answer: 1/4

Step-by-step explanation:

Answer:

1/4

Step-by-step explanation:

We know that 25% is equal to the decimal 0.25. So, we can convert the decimal into a fraction

To do so, we need to write the fraction 0.25 x 100 / 100.

Solving the top part (100 x 0.25), we get 25/100.

Next, we need to simplify this fraction. The GCF (greatest common factor) of both numbers is 25, so we divide 25 by 25 and 100 by 25.

This leaves us with 1/4, and our answer.

Get the two triangles to face in the same direction, or in the same orientation, to check if they resemble one another.

Three sides and three vertices make up a triangle, which is a polygon. It is among the fundamental forms in geometry. Triangle ABC is a graphical representation of a triangle having vertices A, B, and C. When the three points are not collinear, a unique plane and triangle in Euclidean geometry are found. A triangle is a three-sided polygon that is also referred to as a trine on occasion (less frequently). There are three sides and three angles in every triangle, some of which may be equal.

Get the two triangles to face in the same direction, or in the same orientation, to check if they resemble one another. You do this by twisting one form such that it is in alignment with the other. A rotation is the name given to such a change.

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The point that represents a solution to the linear equation 5x + 6y = -15 is (-1, -5 / 3)

Linear equation can be represented in different form such as standard form, general form, slope intercept form and point slope form.

Therefore, let's find the solution to the linear equation.

5x + 6y = -15

Let's put the equation in slope intercept form.

5x + 6y = -15

6y = -5x - 15

divide through by 6

y = - 5 / 6 x - 15 / 6

y = - 5 / 6 x - 5 / 2

Therefore, the slope of the equation is -5 / 6 and the y-intercept is - 5 / 2.

Using the equation we can find the solution of the equation.

We will substitute the value of x to get y value.

Hence,

y = - 5 / 6 x - 5 / 2

using (-1, -5 / 3)

y = - 5 / 6 (-1) - 5  /2

y = 5 / 6 - 5 / 2

y = - 5 / 3

Hence, the solution is (-1, -5 / 3)

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The minimum number of unbroken briquets the bag would have to contain for the company to avoid having to replace the bag for free is b) 411.

A standard deviation (or σ) is a degree of the way dispersed the records is with regards to the mean. Low standard deviation approach records are clustered across the mean, and excessive widespread deviation suggests records are greater unfold out. The interval of filled bags is at confidence interval of 95%. At 95% confidence interval, Z=1.96 (from Z tables).But, Z= (x-mean)/SD => 1.96 = (x-450)/20 => x (upper limit)= 20*1.96+450489 bags. Lower limit or minimum number bags to avoid replace= 450-1.96*20 =411 bags.

Therefore, the correct option is b.

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a) The probability the home team wins is of: 0.7291 = 72.91%.

b) The conditional probability that the home team wins, given that is is behind 5 points by the end of halftime is of: 0.3409 = 34.09%.

c) The conditional probability that the home team wins, given that it is ahead by 5 points at the end of the first quarter is of: 0.9015 = 90.15%.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The mean and the standard deviation for each quarter are given as follows:

Hence for each case, the mean and the standard deviation are given as follows:

The probability of the home team winning is the probability of X being greater than zero for four quarters, hence it is one subtracted by the p-value of Z when X = 0, hence:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

Z = (0 - 6)/9.8

Z = -0.61

Z = -0.61 has a p-value of 0.2709.

1 - 0.2709 = 0.7291 = 72.91%.

When the home team is down five points at the end of the halftime, the probability of winning is P(X > 5) for two quarters, hence one subtracted by the p-value of Z when X = 5 as follows:

Z = (5 - 3)/4.9

Z = 0.41

Z = 0.41 has a p-value of 0.6591.

Z =  has a p-value of 0.6591.

1 - 0.6591 = 0.3409 = 34.09%.

When the home team is up five points at the end of the first quarter, the probability of winning is P(X > -5) for three quarters, hence one subtracted by the p-value of Z when X = -5.

Z = (-5 - 4.5)/7.35

Z = -1.29

Z = -1.29 has a p-value of 0.0985.

1 - 0.0985 = 0.9015 = 90.15%.

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Answer:

[tex]m\angle DAC=35^{\circ}[/tex]

Step-by-step explanation:

Arc [tex]BC=50^{\circ}[/tex] (inscribed angle theorem)

Arc [tex]AB=130^{\circ}[/tex] (semicircle)

[tex]m\angle ADB=65^{\circ}[/tex] (inscribed angle theorem)

[tex]m\angle DEA=80^{\circ}[/tex] (linear pair)

[tex]m\angle DAC=35^{\circ}[/tex] (angle sum of a triangle)

The kangaroo must take at least 23 leaps to catch up to the rabbit.

To catch up to the rabbit, the kangaroo must travel a total distance of 150 feet. For every 12-foot leap of the kangaroo, the rabbit leaps 7 feet, so the kangaroo must take one more leap than the rabbit in order to cover the same distance.

The number of leaps the rabbit takes can be found by dividing the distance the rabbit travels (150 feet) by the distance the rabbit leaps in each jump (7 feet):

150 feet / 7 feet/leap = 21.43 leaps

Since the kangaroo must take one more leap than the rabbit, the kangaroo must take 21.43 + 1 = 22.43 leaps to catch up to the rabbit.

However, since the number of leaps must be a whole number, the kangaroo must take at least 23 leaps to catch up to the rabbit.

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The number of leaps the rabbit takes can be found by dividing the distance the rabbit travels (150 feet) by the distance the rabbit leaps in each jump (7 feet):

150 feet / 7 feet/leap = 21.43 leaps

Since the kangaroo must take one more leap than the rabbit, the kangaroo must take 21.43 + 1 = 22.43 leaps to catch up to the rabbit.

Answer: false

Step-by-step explanation:

The exact length of the curve of the equation y = 1/4 x² − 1/2 ln(x), 1 ≤ x ≤ 4 is 1.125

Here we have given an equation y = 1/4 x² − 1/2 ln(x) with the limit as 1 ≤ x ≤ 4.

Now, the arc length is calculated by differentiating the terms, then we get,

=> dy/dx = x/2 - 1/2x

Now, we have to apply these derivative on the arc length formula, then we get,

=> length =  [tex]\int\limits^4_1 {\sqrt{1 + (\frac{x}{2} -\frac{1}{2x})^2 } } \, dx[/tex]

When we simplify this one then we get,

=> length = [tex]\frac{1}{2} \int\limits^4_1 {\sqrt{x^2 + 2 + \frac{1}{x^2} } } \, dx[/tex]

When we further simplify this one then we get,

=> length = [tex]\frac{1}{2} \int\limits^4_1 {x + \frac{1}{x} } \, dx[/tex]

Now, the result of the differentiation is

=> length = 1/2 [17/4 - 2]

=> length = 1/2 [9/4]

=> length = 1.125

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Yes, y = -3x is proportional because the ratios are equivalent, the equation is proportional.

A mathematical comparison of two numbers is called a proportion. According to proportion, two sets of provided numbers are said to be directly proportional to one another if they increase or decrease in the same ratio. '::' or '=' are used to indicate proportions. Using 2:5::4:8 as an example, 2/5 Equals 4/8 Here, the averages are 5 and 4 and the extremes are 2 and 8. A proportion is an equation that sets two ratios at the same value.

Given

y = -3x

Yes, Because the ratios are equivalent, the equation is proportional.

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Therefore , the algebra is derived from Islam word  Arabic al‐jab used by its  founder.

Mathematical branch known as algebra deals with symbols and the rules governing their manipulation. These characters, which are now expressed as Latin and Greek letters, stand for variables, or quantities without set values, in elementary mathematics. Similar to how sentences illustrate the connections between particular words, in algebra,

Here,

The word algebra is derived from the Arabic word al‐jabr, which its creator, Muhammad ibn Mūsā al‐Khwārizmī, used in the title of his work.

Al-Jabr wa-l-muqbalah, which dates to the ninth century (The Science of Equations and Balancing).

Therefore , the algebra is derived from Islam word  Arabic al‐jab used by its  founder.

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When constructing an angle bisector, the compass should be used to make three arcs, but these arcs do not necessarily need to have the same radius.

To construct an angle bisector, you can follow these steps:

In step 1, the radius of the first arc can be any length, as long as it intersects both sides of the angle. The radius of the second and third arcs should be the same as the first arc because they are drawn using the same width setting on the compass. However, the radius of the second and third arcs does not need to be the same as the radius of the first arc. The important thing is that all three arcs are drawn using the same width setting on the compass so that they are all equidistant from the vertex of the angle.

The complete question is:-

When constructing an angle bisector, the compass must be used to make three arcs. Do all three arcs need to have the same radius? Explain.

- Yes, because then the marks are equidistant from each other.

- Yes, because you need the points of intersection of the angle and arcs to form a parallelogram.

- No; the second two arcs must have the same radius but it can be different from the first.

- No; any three radii will work.

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In a list or collection of numbers, the range is the difference between the lowest and highest integers. Place all the numbers in order before attempting to determine the range.

The statistics' range is the difference between the highest and lowest values for a specific data collection. The range could also be expressed as the difference between the highest and lowest observation. The sample interval is calculated as the greatest value less the smallest value. The sample range is a crucial measure of variability for continuous variables.

here

The x values for a function like f are contained in this collection (x).

The collection of numbers that the function assumes is known as its range.

Following the insertion of an x value, the function outputs this sequence of values.

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Based on the graph, a solution for the system of inequalities is 70 and 30.

In order to write a system of linear inequalities to describe this situation, we would assign variables to the number of 2 pounds tickets sold and number of 3 pounds tickets sold, and then translate the word problem into algebraic equation as follows:

Since less than 100 people attend an exhibition, we have;

x + y < 100           ..........equation 1.

Additionally, more than €260 is raised with tickets of €2 and €4;

2x + 4y > 260         ..........equation 2.

Next, we would use an online graphing calculator to graphically solve the system of linear inequalities. Hence, the required solution for the given system of linear inequalities is the point of intersection of the two (2) lines on the graph, which is given by the ordered pair (70, 30)

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Complete Question:

Less than 100 people attend an exhibition and more than €260 is raised with tickets of €2 and €4. How many tickets of each type could have been sold?

Answer:

Step-by-step explanation: 62 is the limit

We determine the value of log₂16 = 4 by using logarithm laws.

What is logarithm?

Logarithm is the alternative way of representing an exponent. The power at which a number is needed to raise to get the other values is termed as logarithm.

It is called log base 2 or binary logarithm. It is the inverse of the power of function 2.

We stated before that log base 2 is an inverse expression of the power 2. let N is a real positive number and x is the number of exponents. Now, logarithm form of the exponent is written as log₂ N = x

                                                                        2ˣ = N

We are given, log₂ 16

 N= 16

  log₂16 = x

    2ˣ =16

we know that 2⁴ =16

hence, x= 4

hence, log₂ 16 = 4

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a. Yes Juan is correct.

b. The given quadratic equation can be solve by using quadratic formula.

What is a quadratic equation?

A quadratic equation is a polynomial equation with a maximum degree of two. Where a ≠ 0, the equation is given by ax² + bx + c = 0.

Given equation is x² + 7x + 3.

The given equation can't factorized by using middle term or splitting method.

Factorization of quadratic equations can be done in different methods. They are:

Splitting the middle term

Using formula

Using Quadratic formula

Using algebraic identities

Assume that x² + 7x + 3 = 0

x = [-7 ± √(7² - 4 × 1 × 3)]/ 2 × 1

x = [-7 ± √37]/ 2

The factors of quadratic polynomial is ( x - [-7 ± √37]/ 2).

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Answer:

The answer is 35.

Step-by-step explanation:

The given polynomial is p(x) = 5x² -4x + 2 we substitute x = 3 in the polynomial:

p(3) = 5(3)² -4(3) + 2 = 5×9 - 12 + 2 = 45 - 12 + 2 = 47 - 12 = 35

Hence, p(3) =35