Write a polynomial in intercept form so that P(x)=0 has the given roots.
7. 2, √√3
8. -i, -5
9. i, √7
11. --√6, √3
10. 2i, -i
12. 4DR, VII

If the derivative of the function is positive, the function is increasing; if the derivative is negative, the function is decreasing.

What is a logarithmic function?

A logarithmic function is a type of mathematical function that expresses the power to which a given number (the base) must be raised in order to produce a certain value. It is typically written in the form: [tex]f(x) = log_b(x),[/tex] where b is the base of the logarithm.

To determine whether a logarithmic function is increasing or decreasing, you can look at the sign of the derivative of the function. If the derivative is positive, the function is increasing; if the derivative is negative, the function is decreasing.

In the case of logarithmic functions, the derivative of [tex]f(x) = log_b(x)[/tex] is f'(x) = 1/x*ln(b), which is always positive for x > 0, so the logarithmic function is increasing for any base b.

Hence, if the derivative of the function is positive, the function is increasing; if the derivative is negative, the function is decreasing.

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If the derivative of the function is positive, the function is increasing; if the derivative is negative, the function is decreasing.

What is a logarithmic function?

A logarithmic function is a type of mathematical function that expresses the power to which a given number (the base) must be raised in order to produce a certain value. It is typically written in the form: [tex]f(x)=log_b(x)[/tex]. where b is the base of the logarithm.

To determine whether a logarithmic function is increasing or decreasing, you can look at the sign of the derivative of the function. If the derivative is positive, the function is increasing; if the derivative is negative, the function is decreasing.

In the case of logarithmic functions, the derivative of [tex]f(x)=log_b(x)[/tex]. is f'(x) = 1/x*ln(b), which is always positive for x > 0, so the logarithmic function is increasing for any base b.

Hence, if the derivative of the function is positive, the function is increasing; if the derivative is negative, the function is decreasing.

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By adding the squares of the triangle's two smaller sides and comparing the result to the square of a largest side, you may determine whether triangle is acute, obtuse, or right.

The equilateral triangle comes first. All three sides of this triangle are the same length, and each angle measures exactly 60 degrees. An equilateral triangle is Triangle A.

This triangle has two acute angles, which are angles smaller than 90 degrees in length, but one right angle (as well as angle which measures 90 degrees). A right triangle is triangle B.

The isosceles triangle is another. This triangle has two sides that are the same length.

The scalene triangle comes next. This triangle has three different lengths for its three sides.

The acute triangle is the next type of triangle and has three acute angles.

The obtuse triangle is the last triangle. Two of the angles of this triangle are sharp, and one of the angles is obtuse, which is defined as being more than 90 degrees.

Thus, by adding the squares of the triangle's two smaller sides and comparing the result to the square of a largest side, you may determine whether triangle is acute, obtuse, or right.

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Answer: John is 29 years old and Jane is 36 years old.

Step-by-step explanation:

There are many parts to this problem, so let's start by defining John as x and Jane as y.

We know that the sum of their age 10 years ago was 55, which gives us x+y-10=55. The reason we put -10 is because of 10 years ago. If you think about it in terms of now, 10 years back is 55, so now they would be 65 years old. So we can have either x+y-10=55 or x+y=65. For simplicity, I will use x+y=65.

We also know that John is 7 years younger than Jane, which gives x=y-7.

Luckily, we have x defined for us, so we can directly substitute that into our equation.

[tex](y-7)+y=65[/tex]         [combine like terms]

[tex]2y-7=65[/tex]                [add both sides by 7]

[tex]2y=72[/tex]                      [divide both sides by 2]

[tex]y=36[/tex]

Now we know that y=36, we can plug that into either equations to find x.

[tex]x+36=65[/tex]                [subtract both sides by 36]

[tex]x=29[/tex]

This concludes that John is 29 years old and Jane is 36 years old.

Answer: 8:15pm

Step-by-step explanation:

9+6=15 +105 = 120, 120/60=2 hours (total duration), 6:15 + 1 hour = 7:15 + 1 hour = 8:15pm

The number of pores that are on the top of your foot is approximately 13,000 pores on the top of your foot.

To find the number of pores on the top of your foot, we can multiply the area of the top of your foot in square inches by the number of pores per square inch.

The area of the top of your foot is 2.0 x 10^1 in^2 and each square inch of your body has about 6.5 x 10^2 pores. So, the number of pores on the top of your foot can be calculated as:

2.0 x 10^1 in^2 * 6.5 x 10^2 pores/in^2 = 130 x 10^2 pores = 13000 pores

So, there are approximately 13,000 pores on the top of your foot.

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Answer:

is not 37.5

it's 25

Step-by-step explanation:

25+25+15+115=180

50+130=180

180=180

Answer:

25°

Step-by-step explanation:

25 + x + 15 = 3x – 10 ( Sum of two opposite interior angles is equal to the exterior angle)

40 + x = 3x - 10

40 + 10 = 3x - x

50 = 2x

[tex]x = \frac{50}{2} [/tex]

x = 25°

Answer:

Step-by-step explanation:

Selling price of two irons is Rs.1050 each.

Loss on one iron = Rs.300

Profit on second iron = Rs.450

We know that ,

Cost price = Selling price + loss ,when loss is given

and cost price = Selling price - profit ,when profit is given.

therefore, cost price of first iron = 1050 + 300 =1350 rupees

cost price of second iron = 1050 - 450 = 600 rupees

total cost price = 1350 + 600 = 1950 rupees

total selling price = 1050 + 1050 = 2100 rupees

since total cost price < selling price

so, there is a profit of 2100 - 1950 = 150 rupees.

The set of ordered pairs represents the function.

A relation between a collection of inputs and outputs is known as a function. A function is a connection between inputs in which each input is connected to precisely one output. Each function has a range, co-domain, and domain.

Given:

The set of ordered pairs:

(7,9), (4,18), (−3,11), (16,18), (4,−11)

Pairs written as input and output of the relation.

In the attached table,

To prove function:

A relation has an input value which corresponds to an output value. When each input value has one and only one output value, that relation is a function.

And here each input is connected to precisely one output.

Therefore, this is a function.

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Answer:

-0.05

Step-by-step explanation:

10x+0.5

10x=-0.5

x=-0.5/10

x=-0.05

As per the concept of graph, the name of the point where the two axes cross is origin.

Graph in math is defined as a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles.

Here we have to identify the name of the point where the two axes cross.

As we all know that the horizontal axis in the coordinate plane is called the x-axis and the vertical axis is called the y-axis.

Here we have also know that the point at which the two axes intersect is called the origin.

Where the origin is at 0 on the x-axis and 0 on the y-axis.

And these intersecting x- and y-axes divide the coordinate plane into four sections

Finally all these four sections are called quadrants.

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The value of line GB is 6

From the diagram shown, we can see that AG, GB and AB form a triangle with two of its sides being equal.

Also, line AG = line GB

Where;

Then, we have;

Line AG = LIne GB

Substitute the expressions, we have;

5x - 24 = 3x - 12

Collect like terms

5x - 3x = - 12 + 24

Add or subtract like terms

2x = 12

Divide both sides by the coefficient of x

x = 12/ 2

x = 6

The, GB = 3(6) - 12 = 6

Hence, the value is 6

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The value of 5 1/7 minus 2 5/21 will be 2 19 / 21.

A fraction simply means a piece of a whole. In this situation, the number is represented as a quotient such that the numerator and denominator are split. In this situation, in a simple fraction, the numerator as well as the denominator are both integers.

Based on the information, the value of 5 1/7 minus 2 5/21 will be:

= 5 1/7 - 2 5 / 21

= 5 3/21 - 2 5 / 21

= 2 19 / 21

The value is 2 19/21.

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The quadratic equations are approximately equals at the points (0.618,4.236) and (-1.618 , -0.236)

What in mathematics is a quadratic equation?

A quadratic equation is a second-order polynomial equation in one variable using the formula x = ax2 + bx + c = 0 and a 0.

                    The fundamental theorem of algebra ensures that it has at least one solution because it is a second-order polynomial problem. 

 we have

 y = x² + 3x + 2   .............(a)

 y = 2x + 3    ...................(b)

To solve the system of equations equate equation A to equation B

  x² + 3x + 2 = 2x + 3

 Group terms that contain the same variable

   x² + 3x + 2 - 2x - 3 = 0

  x² + x - 1 = 0

The formula to solve a quadratic equation of the form ax² + bx+ c  is equal to

     in this problem we have

       x² + x - 1 =0

 so,   a= 1

        b = 1

        c = -1

substitute in the formula

  x = -1 ± √1² - 4(1)(-1)/2(1)

  x = -1 ±√5/2

  x = -1+√5/2 = 0.618

   x = -1 - √5/2  =  - 1.618

  Find the values of y

 for  x = 0.618

    y = 2(0.618) + 3 = 4.236

  for  x = - 1.618

   y = 2(-1.618) + 3  = -0.236

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The complete question is -

At which points are the equations y = x2 + 3x + 2 and y = 2x + 3 approximately equal?

Answer:

2^x=1

x=0

32•x=0

32.0=0

maybe

[tex](\stackrel{x_1}{3}~,~\stackrel{y_1}{-4})\hspace{10em} \stackrel{slope}{m} ~=~ - 6 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-4)}=\stackrel{m}{- 6}(x-\stackrel{x_1}{3}) \implies y +4= -6 (x -3) \\\\\\ y+4=-6x+18\implies {\Large \begin{array}{llll} y=-6x+14 \end{array}}[/tex]

let's reword this a bit differently, since it looks jumbled

let's find the equation of a line that is perpendicular to y = -5x + 5, and it also passes through (10 , 7), well, keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above

[tex]y=\stackrel{\stackrel{m}{\downarrow }}{-5}x+5\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ -5 \implies \cfrac{-5}{1}} ~\hfill \stackrel{reciprocal}{\cfrac{1}{-5}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{1}{-5} \implies \cfrac{1}{ 5 }}}[/tex]

so we're really looking for the equation of a line with a slope of 1/5 and that it passes through (10 , 7)

[tex](\stackrel{x_1}{10}~,~\stackrel{y_1}{7})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{1}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{7}=\stackrel{m}{ \cfrac{1}{5}}(x-\stackrel{x_1}{10}) \\\\\\ y-7=\cfrac{1}{5}x-2\implies {\Large \begin{array}{llll} y=\cfrac{1}{5}x+5 \end{array}}[/tex]

Therefore , the solution of the given problem of arithmetic mean comes out to be a(30) = 265.

When all of the values in a set of data have the same unit of measurement, such as when all of the numbers are heights, miles, hours, etc., this technique is utilized. Take the numbers 4, 7, 9, and 10 as an illustration. The count of numerals is 4, and the sum of the numbers is 30. 30 divided by 4 equals 7.5, which is the numbers' arithmetic mean.

Here,

Given that there is a common difference between each word, this is an arithmetic sequence. In this instance, the following term is obtained by adding 9 to the phrase before it in the sequence.

Alternatively put,

=> an=a1+d(n−1)

.Arithmetic Sequence:

d=9

This is the formula of an arithmetic sequence.

=>an=a1+d(n−1)

Substitute in the values of

=> a1 =4 and

d=9

.=>an=4+9(n−1)

Simplify each term.

Tap for more steps...

an=4+9n−9

Subtract 9 from 4

an=9n−5

Thus 30th term :

=> a(30)=9(30)−5

=> a(30) = 265

Therefore , the solution of the given problem of arithmetic mean comes out to be a(30) = 265.

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In linear equation, 35 feet per minute is the rate of descent per minute.

What in mathematics is a linear equation?

A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept.        

                         Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

It’s 35 feet per minute. 175 divided by 5 equals 35.

   A submarine descends =  175 ft

                   Time = 5 minutes

         The rate of descent is constant = 175/5

                                             = 35 feet per minute

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Answer:

x = -4

y = 29

Step-by-step explanation:

Let's take our two equations and turn the decimals into fractions for simplicity's sake. We get:

x/4 + 5y/100 = 9    x + y = 25

Simplifying a bit gives us:

x/4 + y/20 = 9       x + y = 25

Now, we will take the first equation and multiply it by 20 to get rid of the fractions. We get:

5x + y = 9

We will rearrange numbers to give us an answer to y that we can substitute in. we get:

y = 9 - 5x

Then, we will plug this into our second equation giving us:

x + 9 - 5x = 25

Combining like terms and subtracting 9 from both sides gives us:

-4x = 16

Dividing by -4 gives us:

x = -4

Then, we can substitute this into our second equation giving us:

-4 + y = 25

We will add 4 to both sides giving us:

y = 29

Hope this helped!

Answer:

[tex]x=-6, 0[/tex]

Step-by-step explanation:

Using the zero-product property, we know that [tex]x=0[/tex] or [tex]x+6=0[/tex].

This means [tex]x=-6, 0[/tex].

The volume of the cone is one third the volume of the cylinder.

The volume of a sphere is 2/3 of the volume of a cylinder with same radius, and height equal to the diameter

Similar to a prism, a cylinder has two bases, but they are circles rather than polygons. A cylinder's sides are also curved rather than flat. A cone has a vertex that is not on the base and a single circular base. The sphere is a figure in space with equal distances between each of its points and the center.

The cone's volume is one-third that of the cylinder. A sphere's volume is 2/3 that of a cylinder with the same radius, and its height is equal to its diameter.

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Answer:

Step-by-step explanation:

-1 divided by x

1 - x

X - 1

Volume of the triangle is 1420m³ given in the question.

There are various volumes for various geometric shapes. Cubes, cuboids, spheres, cylinders, cones, etc. are examples of common 3D shapes that we encounter on a daily basis.

Finding the volume is simple when dealing with shapes with a flat surface, like a cube or a cuboid. Cones, cylinders, and spheres are examples of curved shapes, and it is necessary to take into account both their curved surfaces' dimensions and their overall curvature. Possibly their diameters or radii

The term "irregular solids" refers to solids that lack precise measurements or shapes. A fixed formula that can be used to calculate the volumes of such solids is typically absent. It is possible to obtain these volumes in a variety of ways, one of which is the liquid displacement method, which is based on the Archimedes Principle.

Now volume triangle = area × height

⇒ 1/2 × 20 × 14.2 × 10

= 1420m³

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The two types of Angles of the triangle are  Acute Angle ( angle that are less than 90 degree) and Obtuse Angle ( angle between 90 and 180 degree ).

What are Angles ?

The Angle is defined as the geometrical shape that is constructed by joining two rays to each other at their end points.

The two types of angles of the triangle are Acute Angle and Obtuse Angle :

(i) Acute Angle :

An angle can be called as Acute if the measure of the angle is less than 90 degree .

(ii) Obtuse Angle :

an angle can be called as Obtuse if the measurement of the angle is greater than 90 degree and less than 180 degree .

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After 2 years, Deshaun will have earned $4,000 * 6% = $240 in interest.

The total amount of money in the account at the end of 2 years will be $4,000 + $240 = $4,240. This is the amount Deshaun will withdraw.

Therefore the correct answer is A. Adam earns $5.25 more per hour than Kim.

A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.

Any number of equal parts is represented by a fraction, which also represents a portion of a whole. A fraction, such as one-half, eight-fifths, or three-quarters, indicates how many components of a particular size there are when stated in ordinary English.

Adam earns $186 for 8 hours of work, so his hourly rate is $186/8 = $23.25 per hour

Kim earns $54 for 3 hours of work, so her hourly rate is $54/3 = $18.00 per hour

Kim earns $125 for 7 hours of work, so her hourly rate is $125/7 = $17.86 per hour

Kim earns $216 for 12 hours of work, so her hourly rate is $216/12 = $18.00 per hour

Adam earns $23.25 per hour and Kim earns $18.00 per hour on average.

Adam earns $23.25 - $18.00 = $5.25 more per hour than Kim.

Therefore the correct answer is A. Adam earns $5.25 more per hour than Kim.

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The coordinates of B are (6,0).

The midpoint of a line segment is the point that is exactly halfway between the two endpoints of the segment. For a line segment with endpoints (x1, y1) and (x2, y2), the midpoint is given by the coordinates:

In this problem, we're given that the midpoint is (2, 4) and we need to find the coordinates of point B. Since we know the coordinates of the midpoint, we can set up the equation:

We know that point A = (-2,9) is one endpoint, so we can substitute it into the equation

Solving for x2 and y2, we get x2 = 6 and y2 = 0 which is the coordinates of point B.

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