Which shows one way to determine the factors of 4x3 + x2 – 8x – 2 by grouping?

Answer:

a relation and function ⇒ answer C

Step-by-step explanation:

* Lets revise the relation and the function

- The relation is between the x-values and y-values of ordered pairs.

- The set of all values of x is called the domain, and the set of all values

 of y is called the range

- The function is a special type of relation where every x has a unique y

- Every function is a relation but not every relation is a function

* Lets solve the problem

∵ The graph is a parabola

∵ The parabola is a function because every x-coordinates of the

   points on the parabola has only one y-coordinate

- Ex: some ordered pairs are (-5 , -5) , (-2 , 5) , (0 , 7) , (2 , 5) , (5 , -5)

∵ Every x-coordinate has only one y-coordinate

∴ The graph represents a function

∵ Every function is a relation

∴ The set of ordered pairs in the graph below can be described as

   a relation and function

Answer:

The vertex of the function is (1 , 2)

The domain is (-∞ , ∞) OR {x : x ∈ R}

The range is [2 , ∞) OR {y : y ≥ 2}

Step-by-step explanation:

* Lets revise the standard form of the quadratic function

- The standard form of the quadratic function is

  f(x) = a(x - h)² + k , where (h , k) is the vertex point

- The domain is the values of x which make the function defined

- The domain of the quadratic function is x ∈ R , where R is the set

 of real numbers

- The range is the set of values that corresponding with the domain

- The range of the quadratic function is y ≥ k if the parabola upward

  and y ≤ k is the parabola is down ward

* Lets solve the problem

∵ f(x) = 3(x - 1)² + 2

∵ f(x) = a(x - h)² + k

∴ a = 3 , h = 1 , k = 2

∵ The vertex of the function is (h , k)

∴ The vertex of the function is (1 , 2)

- The domain is all the real number

∵ The domain of the quadratic function is x ∈ R

∴ The domain is (-∞ , ∞) OR {x : x ∈ R}

- The leading  coefficient of the function is a

∵ a = 3 ⇒ positive value

∴ The parabola is opens upward

∴ The range of the function is y ≥ k

∵ The value of k is 2

∴ The range is [2 , ∞) OR {y : y ≥ 2}

[tex]\bf \begin{array}{lll} term&value\\ \cline{1-2} a_1&128\\ a_2&128r\\ a_3&128rr\\ &128r^2 \end{array}~\hspace{5em}\stackrel{a_3}{128r^2}=\stackrel{a_3}{8}\implies r^2=\cfrac{8}{128}\implies r^2=\cfrac{1}{16} \\\\\\ r=\sqrt{\cfrac{1}{16}}\implies r=\cfrac{\sqrt{1}}{\sqrt{16}}\implies r=\cfrac{1}{4}\qquad \leftarrow \textit{common ratio} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf n^{th}\textit{ term of a geometric sequence} \\\\ a_n=a_1\cdot r^{n-1}\qquad \begin{cases} a_n=n^{th}\ term\\ n=\textit{term position}\\ a_1=\textit{first term}\\ r=\textit{common ratio}\\ \cline{1-1} n=7\\ a_1=128\\ r=\frac{1}{4} \end{cases}\implies a_7=128\left( \frac{1}{4} \right)^{7-1} \\\\\\ a_7=128\left( \frac{1}{4} \right)^6\implies a_7=128\cdot \cfrac{1}{4096}\implies a_7=\cfrac{128}{4096}\implies a_7=\cfrac{1}{32}[/tex]

Answer:

PQ must bisect AB

Step-by-step explanation:

Answer:

m<SYD = 106.02°. Reason: alternate exterior angles are equal.

Step-by-step explanation:

Answer:

m<SYD = 106.02°. Reason: alternate exterior angles are equal.

Step-by-step explanation:

Answer:

78

Step-by-step explanation:

The given functions are:

[tex]f(x) = - 2x - 7[/tex]

and

[tex]g(x) = - 4x + 6[/tex]

[tex](f \times g)(x) = f(x) \times g(x)[/tex]

[tex](f \times g)(x) = ( - 2x - 7)( - 4x + 6)[/tex]

When we plug in x=-5, we get:

[tex](f \times g)( - 5) = ( - 4 \times - 5 + 6)( - 2 \times - 5 - 7)[/tex]

[tex](f \times g)( - 5) = ( 20 + 6)( 10 - 7)[/tex]

[tex](f \times g)(5) = ( 26)( 3) =7 8[/tex]

Answer:

See explanation

Step-by-step explanation:

Some important information is missing in the question, however I will try to help.

Let us assume the theater is located at (-5,6) and the library is located at (4,1), then we can use the distance formula to find the distance from the Theater to the Library.

The distance formula is given by:

[tex]d = \sqrt{(x_2-x_1) ^{2} +(y_2-y_1) ^{2} } [/tex]

We plug in the values to get:

[tex]d = \sqrt{ {(4 - - 5)}^{2} + {(6 - 1)}^{2} } [/tex]

[tex]d = \sqrt{81 + 25} [/tex]

[tex]d = \sqrt{144} = 12[/tex]

You can plug in the points you have to get the required answer

Answer:

D, √74

Step-by-step explanation:

got it right on odyssey ware

Answer:

Slope = -1

Step-by-step explanation:

Use the following formula:

slope (m) = (y₂ - y₁)/(x₂ - x₁)

Let:

(x₁ , y₁) = (1 , -3)

(x₂ , y₂) = (3 , -5)

Plug in the corresponding numbers to the corresponding variables. Simplify:

m = (-5 - (-3))/(3 - 1)

m = (-5 + 3)/(3 - 1)

m = -2/2

m = -1

The slope of the line is -1.

~

For this case we have that by definition, the slope of a line is given by:

[tex]m = \frac {y_ {2} -y_ {1}} {x_ {2} -x_ {1}}[/tex]

We have as data the following points:

[tex](x_ {1}, y_ {1}) :( 1, -3)\\(x_ {2}, y_ {2}): (3, -5)[/tex]

Substituting the values:

[tex]m = \frac {-5 - (- 3)} {3-1}\\m = \frac {-5 + 3} {3-1}\\m = \frac {-2} {2}\\m = -1[/tex]

Thus, the slope is -1.

Answer:

The slope is -1

The principles of circle geometry dictate that: if chords are equal then their corresponding arcs are equal; if arcs are equal then their chords are equal; and diameters perpendicular to chords bisect the chord. However, some of the proposed arguments in the question are not matching these principles.

The three statements highlighted in this question are the principles of circle geometry.

1. If chords are equal, then arcs are equal: This theorem states that if two chords in a circle are equal in length, then their corresponding arcs (the part of the circumference that the chord subtends) are also equal. If BC = DE as stated, then Arc BC = Arc DE.

2. If arcs are equal, then chords are equal: This is the converse of the first theorem. If two arcs of a circle are equal, then the chords subtending these arcs are equal. However, this principle is not relevant for the condition if AX is perpendicular to BC, then BX = XC.

3. Diameters perpendicular to chords bisect the chord: This theorem states that if a diameter of a circle is perpendicular to a chord, then it bisects the chord. Therefore the statement if Arc BC = Arc DE, then BC = DE is not relevant to this theorem.

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Answer:

5 units

Step-by-step explanation:

According to the given statement  Δ XYZ is translated 4 units up and 3 units left to yield ΔX'Y'Z' which means that each point in ΔXYZ is moved 4 units up and moved 3 units left.

To find the distance of each corresponding point we will use the Pythagorean theorem which states that the square of the length of the Pythagorean of a right triangle is equal to the sum of the squares of the length of other legs

The square of the required distance = 4^2+3^2 = 16+9 =25

By taking root of 25 we get:

√25 = 5

Thus, we can conclude that the the distance between any two corresponding points on ΔXYZ and ΔX′Y′Z′  is 5 units. ..

Answer:

35 books

Step-by-step explanation:

56/8 = 7

7 x 5 = 35

5/8 of 56 = 35

J.K. Rowling will autograph 35 of the 56 books sold, which is calculated by multiplying the total number of books by 5/8.

To find out how many books J.K. Rowling will autograph, we need to calculate 5/8 of the 56 books sold. Here's the step-by-step calculation:

Find out what 5/8 of the total amount is by multiplying the total number of books (56) by 5/8.

To do this, first multiply 56 by 5, which equals 280.

Then divide that number by 8, which equals 35.

Therefore, J.K. Rowling will autograph 35 books.

Answer

Both graphs have the same number of x-intercepts. The graph of the function f(t) = t² has one x-intercept, which is the value of t for which f(t) = t² = 0, and that is t = 0. The graph of the function g(t) = (t - 8)² has also one x-intercept, which is the value of t for which g(t) = 0 and that is t = 8.

The function f(t) = t² is the most simple form of a parabola, so it is considered the parent function. The function g(t) = (t - 8)² is a daughter function of f(t); then, the graph of g(t) is a horizontal translation of the graph of f(t),  8 units to the right, so the number of x-intercepts (the points where the x-axis is crossed or touched by the graph) does not change, it is just their position what changes.

Explanation:

The x-intercepts are the points where the graph of the function touches or crosses the x-axis. They are found by doing the function equal to zero. In this case f(t) = 0 and g(t) = 0.

You can solve easily f(t) = t², as, just by simple inspection, the soluton is t = 0.

Then, when you realize that the function g(t) = (t - 8)² is a horizontal translation (8 units to the right) of the parent function f(t), you can conclude quickly that the number of x-intercepts of both graphs is the same. Thus, uisng the transformation of the parent function, 8 units to the right, you conclude that both the graph of f(t) and the graph of g(t) have the same number of x-intercepts: one.

f rests at (0/0) and grows upwards, so there is only a single x-intercept. g is f moved 3 to the left, so it also only has one intercept but at (-3,0).

Answer:

x can be any value

Step-by-step explanation:

(5^0)^x = 1,

5^0 =1

1^x =1

X can be any value

For this case we have the following expression:

[tex](5^{ 0})^{x} = 1[/tex]

By definition of power properties we have to:

[tex](a ^ n) ^ m = a ^{n * m}[/tex]

Then, rewriting the expression:

[tex]5 ^ 0 = 1[/tex]

Thus, the variable "x" can take any value:

Answer:

The variable "x" can take any value

Answer:

The correct answer option is: infinitely many solutions.

Step-by-step explanation:

We are given the following system of equations:

[tex] 3 x - 2 y = 6 [/tex] --- (1)

[tex] 6 x - 4 y = 1 2 [/tex] --- (2)

Dividing equation (2) by 2 to get:

[tex] 3 x - 2 y = 6 [/tex] --- (3)

As the equation (3) is same as the equation (1), therefore the system will have infinitely many solutions.

Answer:

infinitely many solutions

Step-by-step explanation:

Note that if you multiply the first equation by 2, you get 6x - 4y = 12, which is exactly the same as the second equation.  These two lines coincide, and so there are infinitely many solutions.

For this case we have that by definition of vertical translations of functions, it is fulfilled:

Assume [tex]k> 0[/tex]:

To graph [tex]y = f (x) + k[/tex], the graph k units is moved up.

To graph [tex]y = f (x) -k[/tex], the graph moves k units down.

In the figure it is observed that:

The original function is [tex]G (x) = x ^ 2[/tex], the function was moved 4 units down. So, the new function is:

[tex]F (x) = x ^ 2-4[/tex]

Answer:

Option B

Answer:

Option B is correct.

Step-by-step explanation:

G(x) = x^2

then f(x) = x^2-4

since the graph is shifted 4 units down because when something is subtracted from the function the graph is shifted down.

that's why f(x) = x^2-4.

Hence Option B is correct.

Answer:

see explanation

Step-by-step explanation:

Given a quadratic equation in standard form

ax² + bx + c = 0 : a ≠ 0, then

The nature of it's roots can be determined by the discriminant

Δ = b² - 4ac

• If b² - 4ac > 0 then roots are real and distinct

• If b² - 4ac = 0 then roots are real and equal

• If b² - 4ac < 0 then roots are not real

For x² - 4x + 4 = 0 ← in standard form

with a = 1, b = - 4, c = 4, then

b² - 4ac = (- 4)² - (4 × 1 × 4) = 16 - 16 = 0

Hence roots are real and equal

This can be shown by solving the equation

x² - 4x + 4 = 0

(x - 2)² = 0

(x - 2)(x - 2) = 0, hence

x - 2 = 0 or x - 2 = 0

x = 2 or x = 2 ← roots are real and equal

Answer:

8/5

Step-by-step explanation:

[tex](g\circ h)(-3)[/tex] means [tex]g(h(-3))[/tex].

Start with the inside first: h(-3).

h(-3) means use the function called h and replace the x with -3.  The expression that is called h is 4-x.

4-x evaluated at x=-3 gives us 4-(-3)=4+3=7.

So the value for h(-3) is 7, or h(-3)=7.

Now this is what we thus far:

[tex](g\circ h)(-3)=g(h(-3))=g(7)[/tex].

g(7) means use the function called g and replace x with 7.   The expression that is called g is (x+1)/(x-2).

(x+1)/(x-2) evaluated at x=7 gives us (7+1)/(7-2)=(8)/(5)=8/5.

This is our final answer:

[tex](g\circ h)(-3)=g(h(-3))=g(7)=\frac{8}{5}[/tex].

Answer:

7

Step-by-step explanation:

You can use the distance formula, but since both points have the same y-coordinate, they lie on a horizontal line. Just find the difference between the x-coordinates and take the absolute value.

distance = |-3 - 4| = |-7| = 7

Answer:

The distance is:

[tex]d=7[/tex]

Step-by-step explanation:

The distance d between two points [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] is calculated using the following formula:

[tex]d=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/tex]

In this case the points are:

A (4,2), B (-3,2).

Then the distance is:

[tex]d=\sqrt{((-3)-4)^2 + (2-2)^2}[/tex]

[tex]d=\sqrt{((-7)^2 + (0)^2}[/tex]

[tex]d=\sqrt{49}[/tex]

[tex]d=7[/tex]

Answer:

[tex]a_n=-7+4(n-1)[/tex]

or

[tex]a_n=-7+(n-1)(4)[/tex]

Step-by-step explanation:

-7,-3,1,5,... is a arithmetic sequence.

Arithmetic sequences have a common difference. That is, it is going up by 4 each time.

When you see arithmetic sequence, you should think linear equation.

The point-slope form of a line is [tex]y-y_1=m(x-x_1)[/tex].

m is the common difference, the slope.

Any they are using the point at x=1 in the point slope form.  So they are using (1,-7).

So let's put this into our point-slope form:

[tex]y-(-7)=4(x-1)[/tex]

[tex]y+7=4(x-1)[/tex]

Subtract 7 on both sides:

[tex]y=-7+4(x-1)[/tex]

So your answer is

[tex]a_n=-7+4(n-1)[/tex]

The explicit formula for the sequence -7, -3, 1, 5,... is an = -7 + (n - 1)(4), which corresponds to an arithmetic sequence with a common difference of 4.

The student is looking for the explicit formula for the sequence -7, -3, 1, 5,.... To find the explicit formula for a linear sequence, we look for a pattern in the increments between successive terms. In this sequence, each term increases by 4 from the previous term (-7 + 4 = -3, -3 + 4 = 1, etc.), so the common difference is 4. Using the formula for an arithmetic sequence, which is an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, n is the term number, and d is the common difference, we can find the explicit formula:

an = -7 + (n - 1)(4)

Since the common difference in the question is positive and we are adding it to the first term which is negative, it is clear that choice A is the correct formula representing the given sequence:

an = -7 + (n - 1)(4)

Answer:

Step-by-step explanation:

Let 'c' be the number of child bikes  c=20

and 'a' be the number of adult bikes. a=6

According to the problem  

The restriction of building time for a week is 4c+6a≤120 hours.........(1)

and the restriction of testing time for a week is 4c+4a≤100 hours..........(2)

Lets check whether company can build c=20 and a=6 bikes in a week by putting these values in (1) and (2).

4c+6a≤120 hours.........(1)

4(20)+6(6)≤120

80+36≤120

116≤120 (true)

4c+4a≤100 hours..........(2)

4(20)+4(6)≤100

80+24≤100

104≤100 (true)

Hence,  the company can build 20 child bikes and 6 adult bikes in the week....

ladders leaning against the wall form triangles. If two of these triangles share the same angle (except the one between wall and floor), they're similar. (The angle between wall and floor is the same, and the sum of all angles in a triangle is 180 degrees)

[tex] \frac{7}{10} = \frac{4}{x} \\ x = \frac{40}{7} [/tex]

The distance the wall should place its base will be 5.71 feet.

Trigonometry is the branch of mathematics that set up a relationship between the sides and angle of the right-angle triangles.

ladders leaning against the wall form triangles. If two of these triangles share the same angle (except the one between the wall and floor), they're similar. (The angle between wall and floor is the same, and the sum of all angles in a triangle is 180 degrees)

The distance will be calculated as below:-

( 7 / 10 ) = ( 4 / x )

x = ( 10 x 4 ) / 7

x = 5.71 feet

Therefore, the distance the wall should place its base will be 5.71 feet.

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Answer:

Option C (-0.922107932)

Step-by-step explanation:

Correlation is a concept which explains a linear relationship between two variables. The correlation constant lies between -1 and 1. 0 lies in the center of the interval. A negative correlation means an inverse relationship, and a positive correlation means a direct relationship. 0 technically means no linear relation between the variables. Further the correlation constant lies from 0, more the strength of the relationship. This means that closer the correlation to 1, stronger the positive relationship, and closer the correlation to -1, stronger the negative relationship. It can be seen that Option C (-0.922107932) is the correlation coefficient which is the largest in terms of the magnitude. Therefore, Option C is the correct choice!!!

Answer:

sin(B)=cos(90-B)

Step-by-step explanation:

sin(B)=cos(90-B) is a co-function identity.

We can also prove it using the difference identity for cosine.

Let's do that:

cos(90-B)

equals

cos(90)cos(B)+sin(90)sin(B)

0cos(B)+1sin(B)

0+sin(B)

sin(B)

Therefore cos(90-B)=sin(B).

Answer:

Step-by-step explanation:

x^2-9 = 0

x^2 = 0+9

x^2=9

Take square root  at both sides:

√x^2 =+/-√9

x =+/- 3

This equation has 2 real solutions, x=3 , x=-3....

Answer:

So the multiplicative rate of change is increased by a factor of 5 per 1 unit increase in x.

Step-by-step explanation:

I think your function is off... but I can look at your ordered pairs.

(1,10)

(2,50)

(3,250),..

As the x increases by 1 the y is being multiply by a factor of 5 each time.

So the multiplicative rate of change is increased by a factor of 5 per 1 unit increase in x.

Answer:

28.71

Step-by-step explanation:

The easiest way I do this is by taking the smaller number (33) changing it to a decimal form of .33, and then multiplying it by 87 to get 28.71

Answer:

20/45 & 21/45

Step-by-step explanation:

Find a common denominator. What you do to the denominator, you do to the numerator. In this case, the smallest denominator is 45.

(4/9)(5/5) = 20/45

(7/15)(3/3) = 21/45

The two fractions you have is:

20/45 for 4/9

21/45 for 7/15

~

Answer:

20/45 for 4/9  and 21/45 for 7/15

Step-by-step explanation:

The least common denominator of 4/9 is 20/45.

The least common denominator of 7/15 is 21/45.

Answer:

The correct option is C.

Step-by-step explanation:

Lets solve each option one by one

A) 16a^2-72a+81

According to whole square formula:

a²-2ab+b² =(a-b)²

We have to take the square root of first and third term of each equation.

a² shows the first term = 16a^2

The square root of 16a^2 is 4a.. because 4 is the number which can be multiplied two times to give 16 and when we multiply a two times it gives us a².

b²  shows the third term = 81

The perfect square of 81 is 9.

2ab shows the middle term.

2ab = 2(4a)(9) = 72a

Thus we can factor it as a perfect square trinomial:

a²-2ab+b² =(a-b)²

16a²-72a+81 =(4a-9)²

B) 169x^2+26xy+y^2

a²+2ab+b² =(a+b)²

The square root of 169x² is 13x

Square root of y² is y

The middle term 26xy =2ab= 2(13x)(y)= 26xy

Thus we can factor it as a perfect square trinomial:

a²+2ab+b² =(a+b)²

169x^2+26xy+y^2 = (13x+y)²

C) x^2-18x-81

We can not factor it as a perfect square trinomial because the third term is negative.

D) 4x^2+4x+1

a²+2ab+b² =(a+b)²

The square root of 4x² is 2x

Square root of 1 is 1

The middle term 4x=2ab=2(2x)(1)= 4x

Thus we can factor it as a perfect square trinomial:

a²+2ab+b² =(a+b)²

4x^2+4x+1 = (2x+1)²

Thus the correct option is C....