Explain how to find the distance between -2 and 3
on a number line.

We need to add 81 as the constant term to the binomial (n² - 18n) so that the resulting trinomial is a perfect square, and factorization of that resulting trinomial will be (n - 9).

As per the question statement, we are provided with a binomial (n² - 18n),

And we are required to add a proper constant to the above mentioned binomial so that the resulting trinomial is a perfect square, and then factorise the resulting trinomial.

To answer question, first we need to know the standard form of expanding a perfect square into a trinomial, which goes as,

[(a - b)² = {(a)² + (b)² - (2 * a * b)]

Here, if we compare our concerned binomial (n² - 18n), or, [(n)² - (2 * n * 9)], with the above mentioned standard expansion of a perfect square into a trinomial, we get, (a = n) and (b = 9), and the (b²) term is missing.

Adding [(b²) = (9²) = 81] which is a constant term, to our concerned binomial (n² - 18n), we get, (n² - 18n + 81),

Or, [(n)² + (9)² + (2 * n * 9)],

Which is a trinomial and matches with the standard expansion of a perfect square into a trinomial, and hence,

[(n² - 18n + 81) = (n - 9)²]

That is, we need to add 81 as the constant term to the binomial (n² - 18n) so that the resulting trinomial is a perfect square, and factorization of that resulting trinomial will be (n - 9).

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The solution of the exponential expression will be a⁹.

Powers can simply be expressed in concise form using exponential expressions. The exponent shows how many times the base has been multiplied.

Given exponential expression is (a⁻¹ x a⁻²)⁻³. The expression will be solved as below,

E = (a⁻¹ x a⁻²)⁻³

The base is the same and the variables are in multiplication with powers then the powers will be added,

E = (a⁻³)⁻³

E = a⁹

Therefore, the solution of the exponential expression will be a⁹.

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Answer:

VA: x = 1, x = -1

HA: y = 0

x-int: (4, 0)

y-int: (0, 4)

Step-by-step explanation:

x-int:

0 = (x-4)/((x-1)(x+1))

0 = (x-4) because the (x-1)(x+1) are multiplied by 0

4 = x

x = 4

(4, 0)

y-int:

y = (0 - 4)/((0 - 1)(0 + 1))

y = (-4)/((-1)(1))

y = (-4)/(-1)

y = 4/1 = 4

(0, 4)

The area of wallpaper required is 0.98 [tex]m^{2}[/tex]

Mensuration is a division of mathematics that studies geometric figure calculation and its parameters such as area, length, volume, lateral surface area, surface area, etc. It outlines the principles of calculation and discusses all the essential equations and properties of various geometric shapes and figures. 

Mensuration is a subject of geometry. Mensuration deals with the size, region, and density of different forms both 2D and 3D. Now, in the introduction to Mensuration, let’s think about 2D and 3D forms and the distinction between them.

d = 0.26m

h = 1.2m

s = pi * d * h

 = 0.98 square meter

the area of wallpaper required is 0.98 [tex]m^{2}[/tex]

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Answer:

0.98 m²

Step-by-step explanation:

I got it right hope this helps.

The angle amount of m is 128⁰

Angle can also refer to the length of a rotation or an angle. The Greek word , which means "crooked, curled," and the English word "ankle," are related words. Both stem from the Proto-Indo-European root  which means "to bend" or "bow," in some way.

A plane angle, according to Euclid, is the inclination to each other of two intersecting lines in a plane that are not perpendicular to one another. Proclus asserts that an angle must be either a quality, a number, or a connection. Eudemus used the first idea, viewing an angle as a departure from a straight line; Carpus of Antioch used the second idea, viewing an angle as the distance or space between intersecting lines; and Euclid used the third idea. Vertical angles, opposing angles, or vertically opposite angles refer to a pair of angles that are directly across from one another and are created by two straight lines meeting to form a "X". Vert is the shorthand for them. opp.

The vertical angle theorem states that angles that are vertically opposing are equal. The assertion shown that since both of a pair of vertical angles are supplementary to both of the neighboring angles, the vertical angles are equal in measure, according to Eudemus of Rhodes' attribution of the proof to Thales of Miletus. When Thales visited Egypt, it is recorded in history[9] that he noticed the Egyptians measuring the vertical angles whenever two intersecting lines were drawn to ensure that they were equal. Thales came to the conclusion that one might establish

4x+6=7x-21

3x=27

x=9

x+x+42+42=360

2x+84=360

x=128⁰

Therefore, the angle amount of m is 128⁰

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The required number is 94.

Let the digit on unit's place be 'x'

Digit on ten's place be 'y'

Therefore,

Original number = 10y + x

Given, the sum of the digits = 11

=> x + y = 11 →(I)

When the digits are reversed, new number = 10x + y

Therefore,

(10y + x) - (10x + y) = 63

10y + x - 10x - y = 63

9y - 9x = 63

=> y - x = 7 →(II)

Adding (I) and (II) we get

2y = 18

y = 9

Substituting the value of y in (I) we get

x + 9 = 11

x = 2

Therefore,

Original number = 10y + x = 10(9) + 4 = 94

The required number is 94.

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There are 102 grams of copper in 12 cm³ of copper.

A unit rate means a rate for one of something.

Given that 1 cm³ of copper have mass of 8.5 grams

Therefore, 12 cm³ = 12*8.5 = 102 grams

Hence, There are 102 grams of copper in 12 cm³ of copper.

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Answer:

(0, 6)

Step-by-step explanation:

Given function:

[tex]y=\log_4(2x+64)+3[/tex]

The y-intercept of a function is the point at which the curve crosses the y-axis, so when x = 0.

Therefore, to find the y-intercept, substitute x = 0 into the function:

[tex]\implies y=\log_4(2(0)+64)+3[/tex]

[tex]\implies y=\log_4(64)+3[/tex]

Rewrite 64 as 4³:

[tex]\implies y=\log_4(4)^3+3[/tex]

[tex]\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax[/tex]

[tex]\implies y=3\log_4(4)+3[/tex]

[tex]\textsf{Apply log law}: \quad \log_aa=1[/tex]

[tex]\implies y=3(1)+3[/tex]

[tex]\implies y=3+3[/tex]

[tex]\implies y=6[/tex]

Therefore, the y-intercept of the given function is (0, 6).

The corresponding sides and angles of both the triangles [pre-image and the image] are congruent

Hence,The corresponding sides and angles of both the triangles [pre-image and the image] are congruent

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The x-intercept of the given graph of the function is (10,0) while the y-intercept is (0,15).

A graph is a diametrical representation of any function between the dependent and independent variables.

The graph is easy to understand the behavior of the graph.

For example y = x² form a parabola now by looking at only the graph we can predict that it has only a positive value irrespective of the interval of x.

As per the given equation,

y = (-3/2)x + 15

At y-intercept,

x = 0 so, y = 0 + 15 = 15 thus, (0,15)

At x-intercept

y = 0

0 = (-3/2)x + 15

x = 10 thus, (10,0)

Hence "The x-intercept of the given graph of the function is (10,0) while the y-intercept is (0,15)".

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For the complete question and graph image attached below,

The simple interest using I = prt if Oliver borrowed $1,500 to purchase a riding lawn mowerat the rate of f 12% for 6 monthsis

The concept that will be used is the concpt of simple interest and thei can be calculates using the simple interst formular which is

I = [(prt)/100]

where p = principal =  $1,500

     I = intereste rate

     t=time

Then this can be expressed as

I= (prt)/100

I =  ($1,500* 6/12 * 12 )/100

=  $90

In conclusion the simple interest is now  $90 which is the second option that is given.

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21 is  missing  length of parallel lines.

What is an example of parallel lines?

Because lines are parallel

so they cut into equal proportion

 then ,         4/10  = x/15

                  4 * 15/10 = x

                     x = 6

length of line  = 15 + 6 = 21

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Answer:

Slope is equal to the change in y y over the change in x x, or rise over run. m = change in y change in x m = change in y change in x. The change in x x is equal to the difference in x-coordinates (also called run ), and the change in y y is equal to the difference in y-coordinates (also called rise ). m = y2 −y1 x2 −x1 m = y 2 - y 1 x 2 - x 1.

He may keep 82 movies on each shelf, using up to 492 movies and leaving 3 movies to retain.

Arithmetic operations is a field of mathematics that studies numbers and the operations on numbers that are helpful in all other disciplines of mathematics. It consists mostly of operations like addition, subtraction, multiplication, and division. Arithmetic Operator is used to conduct mathematical operations on the supplied operands such as addition, subtraction, multiplication, division, modulus, and so on. Arithmetic operators carry out operations on numbers. Variables are assigned values using assignment operators. Logical operators compare two values and return either "true" or "false" depending on whether the comparison is true (or false).

Here,

Movies that movie store has=495 movies

Number of shelves=6

If the owner wanted to make sure each shelf had the same number of movies,

=495/6

=82.5

He can keep 82 movies in each shelf that will use up to 492 movies and 3 movies will be left to keep.

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Answer: always, always

Step-by-step explanation:

Every circle is similar to every other circle.

The solution of the algebraic equation is x = -2. The proof is elaborated below.

The algebraic equation given to us is 14(x + 1) = -7(4 + x). An equation is a mathematical statement that proves the equality of two mathematical expressions. The first step is to simplify the brackets. The equation becomes 14x + 14 = -28 - 7x. Now, we will add 7x to both sides of the equation. The equation is modified as 21x + 14 = -28. Now, we will subtract 14 from both sides of the equation. The equation is modified as 21x = -42. Now, we will divide both sides of the equation by 21. The equation is modified as x = -2.

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Answer: 120

explanation: We already have our first value 36 and the second value 30. Let's assume the unknown value is Y which answer we will find out.

As we have all the required values we need, Now we can put them in a simple mathematical formula as below:

STEP 136 = 30% × Y

STEP 236 =

30

100

× Y

Multiplying both sides by 100 and dividing both sides of the equation by 30 we will arrive at:

STEP 3Y = 36 ×

100

30

STEP 4Y = 36 × 100 ÷ 30

STEP 5Y = 120

Finally, we have found the value of Y which is 120 and that is our answer.

You can easily calculate 36 is 30 percent of what number by using any regular calculator, simply enter 36 × 100 ÷ 30 and you will get your answer which is 120

The sum of a number w is more than -12 becomes w > -12

If the equation or inequality contains variable terms, then there might be some values of those variables for which that equation or inequality might be true. Such values are called solution to that equation or inequality.

We have been given that the sum of a number w is more than -12.

Now the sum of a number = w

It is more than -12 means the inequality form as;

w > -12

Therefore, the overall expression becomes w > -12

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The inequality expression of the volume of the balloon is V < 4,400

From the question, we have the following statement:

The initial volume of the balloon is 4,400 cubic centimeters

Also, we understand that:

The balloon leaks out over time

This means that the volume of air in the balloon decreases over time.

So, we have the following representation

V = Less than the initial volume

Rewrite as

V < 4400

Hence, the volume of the balloon at any time after it starts leaking is V < 4,400

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The solution is Pack #1

The pack which offers Toby the best deal is Pack #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 price per card of each pack be = A

Now , the equation is

a)

Pack #1

The number of cards = 12 cards

The cost per pack = $ 29

The equation is

So , the cost of 1 card = cost per pack / number of cards

                                    = $ 2.416

Therefore , the cost of one card in pack 1 is $ 2.416

b)

Pack #2

The number of cards = 24 cards

The cost per pack = $ 89

The equation is

So , the cost of 1 card = cost per pack / number of cards

                                    = $ 3.708

Therefore , the cost of one card in pack 2 is $ 3.708

c)

Pack #3

The number of cards = 24 cards

The cost per pack = $ 213

The equation is

So , the cost of 1 card = cost per pack / number of cards

                                    = $ 8.875

Therefore , the cost of one card in pack 3 is $ 8.875

And ,

cost of one card in pack 3 is $ 8.875 > cost of one card in pack 2 is $ 3.708 > cost of one card in pack 1 is $ 2.416

Therefore , the cost of one card in pack 1 is $ 2.416 and this is the best deal

Hence , The pack which offers Toby the best deal is Pack #1

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The solution to the points are:

This is the point where the curve or curves of the graph changes direction.

In this case, the turning point in the vertex

The graph has its vertex at (1, -3.2)

This means that

Turning point = (1, -3.2)

This is the point where the curve or curves of the graph touches the x-axis i.e. f(x) = 0

In this case, the points are: (-1, 0) and (3, 0)

This means that

The roots are (-1, 0) and (3, 0)

This is the point where the curve or curves of the graph passes through x = 2.5

In this case, the points are: (2.5, -1.4)

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a) a/5 + 3 = 8

Step 1 : Minus both sides by 3

a/5 + 3 - 3 = 8 - 3 -> a/5 = 5

Step 2 : Multiply both sides by 5

a/5 x 5 = 5 x 5 -> a = 25.

b) (3b)/7 - 1 = 5

Step 1 : Add both sides by 1

(3b)/y - 1 +1 = 5 + 1 -> (3b)/7 = 6

Step 2 : Multiply both sides by 7

(3b)/7 x 7 = 6 x 7 -> 3b = 42

Step 3 : Divide both sides by 3

3b : 3 = 42 : 3 -> b = 14.

The solutions to the given equation are -5/2 and  -3. These answer can get after using quadratic formula.

The quadratic formula is a formula that offers a quadratic equation's solution. In addition to the quadratic formula, other methods of solving quadratic equations include factoring, completing the square, graphing, and others.

In the literature, there are numerous ways to derive the quadratic formula. The common one uses the completing the square method in a straightforward manner.

The given equation is :

-11x = 2x² +15

Move terms to the left side

-11x -(2x^2 +15) =0

Rearrange terms

-2x^2 - 11x -15 = 0

Common factor

-(2x^2+ 11x +15) =0

Divide both sides by the same factor

2x^2 +11x +15 =0

Use the quadratic formula

x = -11 ± √11² - 4 * 2*15/2*2

x =-11±1/4

Solve

x = -5/2

x= -3

The value of x will be -5/2 or -3.

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Answer:

y = -2x + 1

Step-by-step explanation:

y = mx + b  you must find the m (slope) and the b (y intercept) to write the equation for the graph.

The slope is negative.  If you start on the very left side of the graph and follow it, you will see that it goes in a down hill direction.  That means that the slope is negative.  If I start on the very left and the line goes up from there, the slope is positive.  If the line is a straight horizontal line, then the slope is zero, it does not go up or down.

The slope is -2.  Select one of the dots and travel to the next dot, but you can only move horizonaly and vertically.  You cannot move diagonally.  You will notice that the two closes dots are two units up (or down) and only 1 unit left (or right) depending on which dot you start at.  Since you move 2 units up and 1 unit left that is a slope of 2.  2/1 = 2.

The b (y-intercept) is just where you see the line cross the y axis.  It crosses it at 1.

Answer:

  y = -2x + 1

Step-by-step explanation:

Here m is the slope and b is y-intercept.

Pick any two points on the line.

(-1,3) & (2,-3).

Find the Slope.

     [tex]\sf \boxed{Slope =\dfrac{y_2-y_1}{x_2-x_1}}[/tex]

                 [tex]\sf = \dfrac{-3-3}{2-[-1]}\\\\=\dfrac{-6}{2+1}\\\\=\dfrac{-6}{3}\\\\=-2[/tex]

m = -2.

Equation of the line:

     y = -2x + b

Substitute any one point in the above equation and find the y-intercept. (-1,3)

     3 = -2*(-1) + b

     3 = 2 + b

3 - 2  = b

      b = 1

Equation of the line:

        y = -2x + 1

The equation which represents a cubic polynomial function with zeroes 2, 3 and 5 is; f(x) = x³ - 10x² + 31x - 30.

It follows that if a polynomial function has zeroes, a, b and c; The factors of such polynomial function are; (x - a), (x - b) and (x - c).

Therefore, since the given polynomial function has zeroes, 2, 3 and 5, the polynomial function is;

f(x) = (x - 2) (x - 3) (x - 5)

Therefore, by simplification; we have;

f(x) = (x² - 5x + 6) (x - 5)

f(x) = x³ - 5x² + 6x - 5x² + 25x - 30.

f(x) = x³ - 10x² + 31x - 30.

Therefore, the required cubic polynomial function is; f(x) = x³ - 10x² + 31x - 30.

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Answer:

x=3 y=4

Step-by-step explanation:

-4x= -6y+12      -1

8x-4y = 8         -2

-4x+6y=12        -1

8x-4y = 8         -2

multiply first equation by 2:

2(-4x+6y=12)

= -8x+12y=24

add with 2nd equation;

-8x+12y=24

       +

8x-4y=8

-8x+8x=0

+12-4y=8y

24+8=32

0+8y=32

y=32/8

y=4

input y into any equation to get x:

-4x= -6(4)+12

-4x = -24+12

-4x= -12

-x= -12/4

x=3

The probability that the number is a multiple of four is 0.25.

we know that if a number is multiple of four then its last 2 digits must be divisible by four.( i.e. ten's and one's places)

We are given only four numbers 1,2,3,4 , which can't be repeated.

So here , in this case , such numbers with last two digits divisible by four are 12,24 and 32 only

Having last two digits ‘12’ ;

the first two digits can be formed using 3 and 4 only . This can be done in (2!) = 2 ways.

Similar case will be happening with 24 and 32 as well.

So the total ways of getting a multiple of four will be = 2 + 2 + 2

                                                                                        = 6 ways

Now total numbers that can be made from given four digits = [tex]P_{4} ^{4}[/tex]

                                                                                                  = 24

So the probability of getting a multiple of four is = 6/ 24

                                                                                 = 0.25

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Answer:

[tex] - 3 \leqslant w < 1[/tex]

Step-by-step explanation:

as shown in picture attached