14. Solve this system of equations:
4x + 2y = 22
3x – 2y = 13

Answer:

f(x) = (x+2)(x+3)(x-(3-6i))(x-(3+6i))

f(x) = 270 + 189 x + 21 x^2 - x^3 + x^4

Step-by-step explanation:

First of all, we must know that complex roots come in conjugate pairs.

So the zeros of your equation would be

x = -2

x = -3

x = 3 - 6i

x = 3 + 6i

Your polynomial is of fourth degree.

f(x) = (x-(-2))(x-(-3))(x-(3-6i))(x-(3+6i))

f(x) = (x+2)(x+3)(x-(3-6i))(x-(3+6i))

Please , see attached image below for full expression

f(x) = 270 + 189 x + 21 x^2 - x^3 + x^4

Answer:

The required polynomial is [tex]P(x)=a\left(x^4-x^3+21x^2+189x+270\right)[/tex].

Step-by-step explanation:

The general form of a polynomial is

[tex]P(x)=a(x-c_1)^{m_1}(x-c_2)^{m_2}...(x-c_n)^{m_n}[/tex]

where, a is a constant, [tex]c_1,c_2,..c_n[/tex] are zeroes with multiplicity [tex]m_1,m_2,..m_n[/tex] respectively.

It is given that  –2, –3,3 – 6i are three zeroes of a polynomial.

According to complex conjugate root theorem, if a+ib is a zero of a polynomial, then a-ib is also the zero of that polynomial.

3 – 6i is a zero. By using complex conjugate root theorem 3+6i is also a zero.

The required polynomial is

[tex]P(x)=a(x-(-2))(x-(-3))(x-(3-6i))(x-(3+6i))[/tex]

[tex]P(x)=a(x+2)(x+3)(x-3+6i)(x-3-6i)[/tex]

[tex]P(x)=a\left(x^2+5x+6\right)\left(x-3+6i\right)\left(x-3-6i\right)[/tex]

On further simplification, we get

[tex]P(x)=a\left(x^3+6ix^2+2x^2+30ix-9x+36i-18\right)\left(x-3-6i\right)[/tex]

[tex]P(x)=a\left(x^4-x^3+21x^2+189x+270\right)[/tex]

Therefore the required polynomial is [tex]P(x)=a\left(x^4-x^3+21x^2+189x+270\right)[/tex].

Answer:

Yes, A is correct.

Answer:

C

Step-by-step explanation:

The general rule for the quadratic function is

[tex]y=ax^2+bx+c[/tex]

Use the data from the table:

[tex]y(50)=130\Rightarrow 130=a\cdot 50^2+b\cdot 50+c\\ \\y(70)=130\Rightarrow 130=a\cdot 70^2+b\cdot 70+c\\ \\y(90)=200\Rightarrow 200=a\cdot 90^2+b\cdot 90+c[/tex]

We get the system of three equations:

[tex]\left\{\begin{array}{l}2500a+50b+c=130\\ \\4900a+70b+c=130\\ \\8100a+90b+c=200\end{array}\right.[/tex]

Subtract these equations:

[tex]\left\{\begin{array}{l}4900a+70b+c-2500a-50b-c=130-130\\ \\8100a+90b+c-2500a-50b-c=200-130\end{array}\right.\Rightarrow \left\{\begin{array}{l}2400a+20b=0\\ \\5600a+40b=70\end{array}\right.[/tex]

From the first equation

[tex]b=-120a[/tex]

Substitute it into the second equation:

[tex]5600a+40\cdot (-120a)=70\Rightarrow 800a=70,\\ \\ a=\dfrac{7}{80},\\ \\ b=-120\cdot \dfrac{7}{80}=-\dfrac{21}{2}=-10.5[/tex]

So,

[tex]2500\cdot \dfrac{7}{80}+50\cdot (-10.5)+c=130\Rightarrow 218.75-525+c=130\\ \\c=130-218.75+525=436.25[/tex]

The quadratic function is

[tex]y=\dfrac{7}{80}x^2-10.5x+436.25\\ \\y=0.0875x^2-10.5x+436.25[/tex]

Answer: the length is 13

Step-by-step explanation:

one side is 26 ft. and y is half of one side so by dividing 26 by 2 you would get 13

Step-by-step explanation:

The x-intercepts are x = -1 and x = 5, so:

y = k (x + 1) (x − 5)

The vertex is (2, -3), so:

-3 = k (2 + 1) (2 − 5)

-3 = -9k

k = 1/3

y = 1/3 (x + 1) (x − 5)

Simplifying:

y = 1/3 (x² − 4x − 5)

y = 1/3 x² − 4/3 x − 5/3

f(x) =  1 / 3 x² - 4 / 3 x - 5 / 3

using the form:

f(x) = a(x - h)² + k

The vertex coordinates are 2 and -3.

h = 2

k = - 3

therefore,

f(x) = a(x - 2)²  - 3

f(x) = a(x - 2)² - 3

let's use the coordinates (-1, 0) to find a. Therefore,

0 = a(-1 - 2)² - 3

0 = 9a - 3

3 = 9a

a = 3 / 9

a = 1 / 3

let's insert the value of a in the equation.

f(x) = a(x - 2)² - 3

f(x) = 1 / 3 ( x - 2)² - 3

f(x) = 1 / 3 (x - 2)(x -2) - 3

f(x) = 1 / 3 (x² - 4x + 4) - 3

f(x) = x² / 3 - 4x / 3 + 4 / 3 - 3

f(x) =  1 / 3 x² - 4 / 3 x - 5 / 3

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Answer:

p(x) = 6(x + 2)² - 3

Step-by-step explanation:

This one requires a lot of thinking because since our A is not 1 and how the quadratic equation looks, we need to think of a low number while "completing the square [½B]²". So, let us choose 2. We set it up like this:

6(x + 2)² → 6(x² + 4x + 4) → 6x² + 24x + 24

6x² + 24x + 24 - 3 → 6x² + 24x + 21 [TA DA!]

We know that our vertex formula is correct. Additionally, -h gives you the OPPOSITE terms of what they really are, and k gives you the EXACT terms of what they really are. Therefore, your vertex is [-2, -3].

I am joyous to assist you anytime.

Answer:

A. 140 cubic inches

Step-by-step explanation:

The total volume is the volume of the rectangular prism at the bottom plus the volume of the pyramid on top.

The volume of the prism is width times length times height.

V = wlh

V = (3)(7)(4)

V = 84

The volume of the pyramid is one third the area of the base times the height.

V = ⅓ Ah

The base of the pyramid is a rectangle.  Its area is the width times length.  The height of the pyramid is the total height minus the height of the prism.

V = ⅓ (3)(7)(12−4)

V = 56

So the total volume is:

V = 84 + 56

V = 140

Answer:

140 cubic inches

Step-by-step explanation:

Answer:

8/7

Step-by-step explanation:

8/7 is an improper fraction.

8 > 7

A fraction has to have the numerator less than the denominator, in order to be a proper fraction.

In this case, 8/7 is the only fraction with a numerator more than the denominator.

Therefore, 8/7 is an improper fraction.

Answer:

8/7 is an improper fraction.

Step-by-step explanation:

An improper fraction is just a fraction where the numerator (top number) is greater than the denominator (bottom number)

2<3

6<11

21<25

8>7

Hope this helps!!!

[tex](f+g)(x)=2x^2+3+x-2=2x^2+x+1\\\\(f+g)(2)=2\cdot2^2+2+1=8+3=11[/tex]

Answer:

11

Step-by-step explanation:

(f+g)(2) means f(2)+g(2).

f(2) means we need to replace the x's in f(x)=2x^2+3 with 2.

This gives us f(2)=2(2)^2+3.

Let's simplify f(2)=2(4)+3=8+3=11.

g(2) means we need to replace the x's in g(x)=x-2 with 2.

This gives us g(2)=2-2

Let's simplify g(2)=0.

Now f(2)+g(2) means we just add the result of f(2) to the result of g(2).

So the problem is what is 11+0?

Answer is 11

Answer:

The midpoint of the line segment is (9,3).

Step-by-step explanation:

To find the midpoint of a line segment, we use the midpoint formula, ( (x1 + x2)/2, (y1+y2)/2).  This means that to find the midpoint, we must add together both of the x-values of the endpoints and divide by 2 and do the same with the y values, basically finding the average of the two endpoints, or the middle.

x1 + x2 = 10 + 8 = 18

18/2 = 9

y1 + y2 = 2 + 4 = 6

6/2 = 3

Therefore, the midpoint of the line segment is (9,3).

Hope this helps!

Answer:

The solution set is the interval (-∞ , 6] OR {x : x ≤ 6}

Step-by-step explanation:

* Lets explain how to find the solution set of the inequality

- The inequality is 5x - 9 ≤ 21

∵ 5x - 9 ≤ 21

- At first add 9 to both sides of the inequality to separate x in one

 side and the numbers in the other sides

∴ 5x - 9 + 9 ≤ 21 + 9

∴ 5x ≤ 30

- Lets divide both sides of the inequality by 5 to find the values of x

∴ (5x ÷ 5) ≤ (30 ÷ 5)

∴ x ≤ 6

- The solutions of the inequality is all real numbers smaller than

  or equal to 6

∴ The solution set is the interval (-∞ , 6] OR {x : x ≤ 6}

- We can represent this inequality graphically to more understand

 for the solution

- From the graph the solution set is the purple area

Answer:

f(x) = (x+2)(x-8)/(x-6)(x+4) <-> x=6,x=-4

i(x) = (x-4)(x-6)/(x-2)(x+8)  <-> x=2,x=-8

k(x) = (x-2)(x+8)/(x+6)(x-4) <-> x=-6,x=4

m(x) = (x+4)(x-6)/(x+2)(x-8)  <-> x=-2,x=8

Step-by-step explanation:

The function is discontinuous if the denominator is zero.

We will check for which function the values are given

1) f(x) = (x+2)(x-8)/(x-6)(x+4)

if x = 6 and x = -4 the  denominator  is zero

So, x=6 and x=-4 given

2) g(x) = (x+4)(x-8)/(x+2)(x-6)

if x = -2 and x = 6 the  denominator  is zero

So, x= -2 and x= 6 not given so, g(x) will not be considered

3) h(x)= (x+2)(x-6)/(x-8)(x+4)

if x = 8 and x = -4 the  denominator  is zero

So, x= 8 and x= -4 not given so, h(x) will not be considered

4) i(x) = (x-4)(x-6)/(x-2)(x+8)

if x = 2 and x = -8 the  denominator  is zero

So, x= 2 and x= -8 given

5) j(x) = (x-2)(x+6)/(x-4)(x+8)

if x = 4 and x = -8 the  denominator  is zero

So, x= 4 and x= -8 not given so, j(x) will not be considered

6) k(x) = (x-2)(x+8)/(x+6)(x-4)

if x = -6 and x = 4 the  denominator  is zero

So, x= -6 and x= 4  given

7) l(x) = (x-4)(x+8)/(x+6)(x-2)

if x = -6 and x = 2 the  denominator  is zero

So, x= -6 and x= 2  not given so, l(x) will not be considered

8) m(x) = (x+4)(x-6)/(x+2)(x-8)

if x = -2 and x = 8 the  denominator  is zero

So, x= -2 and x= 8 given

Answer:

Step-by-step explanation:

[tex]5+20\cdot2^{2-3x}=10\cdot2^{-2x}+5\qquad\text{subtract 5 from both sides}\\\\20\cdot2^{2-3x}=10\cdot2^{-2x}\qquad\text{divide both sides by 10}\\\\2\cdot2^{2-3x}=2^{-2x}\qquad\text{use}\ a^n\cdot a^m=a^{n+m}\\\\2^{1+2-3x}=2^{-2x}\\\\2^{3-3x}=2^{-2x}\iff3-3x=-2x\qquad\text{subteact 3 from both sides}\\\\-3x=-2x-3\qquad\tex\qquad\text{add}\ 2x\ \text{to both sides}\\\\-x=-3\qquad\text{change the signs}\\\\x=3[/tex]

Answer:

The domain is (2,8,12)

Step-by-step explanation:

we have

[tex]f(x)=2x-4[/tex]

we know that

The range is (0,12,20)

Find the domain

1) For f(x)=0 -----> Find the value of x

substitute the value of f(x) in the equation and solve for x

[tex]0=2x-4[/tex]

[tex]2x=4[/tex]

[tex]x=2[/tex]

therefore

For f(x)=0 the domain is x=2

2) For f(x)=12 -----> Find the value of x

substitute the value of f(x) in the equation and solve for x

[tex]12=2x-4[/tex]

[tex]2x=12+4[/tex]

[tex]x=8[/tex]

therefore

For f(x)=12 the domain is x=8

3) For f(x)=20 -----> Find the value of x

substitute the value of f(x) in the equation and solve for x

[tex]20=2x-4[/tex]

[tex]2x=20+4[/tex]

[tex]x=12[/tex]

therefore

For f(x)=20 the domain is x=12

therefore

The domain is (2,8,12)

John's present age is 15 years, and his sister's present age is 10 years.

We have,

Let's denote John's sister's age as x.

According to the given information,

John is five years older than his sister, so John's age would be x + 5.

The product of their present ages is 150, which means:

x * (x + 5) = 150

Expanding the equation:

x² + 5x = 150

Rearranging the equation to standard quadratic form:

x² + 5x - 150 = 0

Now we can solve this quadratic equation.

Factoring or using the quadratic formula will give us the values of x, representing the sister's age.

Factoring the equation:

(x - 10)(x + 15) = 0

Setting each factor to zero:

x - 10 = 0 or x + 15 = 0

Solving for x:

x = 10 or x = -15

Since ages cannot be negative, we can discard the solution x = -15. Therefore, the sister's present age is x = 10.

John's present age is then calculated by adding 5 years to his sister's age:

John's age = x + 5 = 10 + 5 = 15 years.

Thus,

John's present age is 15 years, and his sister's present age is 10 years.

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By forming a quadratic equation from the given problem, we determine that John's present age is 15 years and his sister's present age is 10 years.

Let's designate John's age as J and his sister's age as S. We've been given two pieces of information: John is five years older than his sister, and the product of their present ages is 150. Mathematically, these can be expressed as:

Substituting the first equation into the second gives us:

(S + 5) * S = 150

Expanding and rearranging, we get a quadratic equation:

S2 + 5S - 150 = 0

To solve for S, we can either factor the quadratic equation or use the quadratic formula. Factoring seems simpler in this case:

(S + 15)(S - 10) = 0

This gives us two possible values for S: -15 and 10. Since ages cannot be negative, we discard S = -15 and keep S = 10. Now we can find John's age:

J = S + 5 = 10 + 5 = 15

Therefore, John's present age is 15 years, and his sister's present age is 10 years.

Answer

Y= 3 × 2 + 6x + 1 = 6 + 6x +1 = 6x + (6+1) = 6x + 7

Answer:

This is a function, so the answer is a plot, you can see it in the attached picture

Step-by-step explanation:

This is a function, the best way to go here is to graph.

First lest find the roots of 3x2+6x+1, you do this by using the quadratic equation

[tex]x_{1} = \frac{-b + \sqrt{b^{2}-4ac }}{2a}\\x_{2} = \frac{-b - \sqrt{b^{2}-4ac }}{2a}[/tex]

Where a=3, b=6, c=1

Using that, you get, x1 = -1 - sqrt(2/3) which is a negative number, and x2=sqrt(2/3) - 1 which is a negative number.

These number represent values that makes the function equals =0

Now if we make x=0, the result is y=1,

We have a better idea of what is to plot and it is shown on the attached picture

Answer:

2ε + 12 [two trillion (twelve zeros)]

Step-by-step explanation:

[1][2][4][5][10][1000][2000][2500] (1 is unnecessary)

If you are talking about the FIRST 8 factors of 10000 [10⁴], then here you are. Multiply all those numbers to get the calculator notation answer above, or two trillion.

**This question is not specific enough, which was why I improvised and thought that this was what you meant. I apologize if this was not what you meant and I also hope this was what you were looking for.

I am joyous to assist you anytime.

Answer:

[tex]10^{32}[/tex].

Step-by-step explanation:

eight factors of 10^4 are

[tex]10^4*10^4*10^4*10^4*10^4*10^4*10^4*10^4 = (10^4)^8 = 10^{4*8}=10^{32}[/tex]

that is one hundred nonillion.

Answer:

9* 3 ^ (x-2)

Step-by-step explanation:

g(x) = 3^x

We know a^ (b) * a^(c) = a^ (b+c)

 9* 3 ^ (x+2) = 3^2 * 3 ^(x+2) = 3^(2+x+2) = 3^x+4  not equal to 3^x

3*(9^(x+2)) = 3*3^2(x+2) = 3^1 * 3^(2x+4) =3^(2x+4+1) = 3^(2x+5) not equal  

9* 3 ^ (x-2) = 3^2 * 3 ^(x-2) = 3^(2+x-2) = 3^x  equal to 3^x    

3*(9^(x-2)) = 3*3^2(x-2) = 3^1 * 3^(2x-4) =3^(2x-4+1) = 3^(2x-3) not equal        

Answer:

Second table.

Step-by-step explanation:

A function has an additive rate of change if there is a constant difference between any two consecutive input and output values.

The additive rate of change is determined using the slope formula,

[tex]m = \frac{y_2-y_1}{x_2-x_1} [/tex]

From the first table we can observe a constant difference of -6 among the y-values and a constant difference of 2 among the x-values.

[tex]m = \frac{ - 9- - 3}{4-2} = - 3[/tex]

For the second table there is a constant difference of 3 among the y-values and a constant difference of 1 among the x-values.

The additive rate of change of this table is

[tex]m = \frac{ - 1 - - 4}{3 - 1} = 3[/tex]

Therefore the second table has an additive rate of change of 3.

Answer: it’s A and E

First and last one

Step-by-step explanation:

Answer:

D. size and shape

Step-by-step explanation:

Triangles are congruent if they have the same size and shape.

The domain of the outer function is all real numbers, because f(x) is an exponential function.

The domain and range of g(x) are all real numbers.

So, the domain of the composition is again all real numbers, because there is no way that an output from g(x) will not be a valid input for f(x).

For this case we have the following functions:

[tex]f (x) = 2 ^ x\\g (x) = x-2[/tex]

We must find [tex](f_ {o} g) (x):[/tex]

By definition we have to:

[tex](f_ {o} g) (x) = f [g (x)][/tex]

So:

[tex](f_ {o} g) (x) = 2 ^ {x-2}[/tex]

By definition, the domain of a function is given by all the values for which the function is defined. Thus, the domain of the composite function is:

In this case, there are no real numbers that make the expression indefinite.

Thus, the domain is given by all the real numbers.

Answer:

All the real numbers

Answer:

$28.

Step-by-step explanation:

p(x) = -2(x - 9)^2 + 100

If the prices (x) = 15 dollars we work out the profit by substituting x = 15 in the above formula:

p(15) = -2(15 - 9)^2 + 100

= -2 * 6^2 + 100

= $28.

Answer:

28

Step-by-step explanation:

Answer: 6 days.

Step-by-step explanation:

Inverse proportion equation has the form:

[tex]y=\frac{k}{x}[/tex]

Where "k" is the constant of proportionality.

Let be "y" the time it takes to do a job (number of days) and "x" the number of workers.

We can find "k" knowing that  it takes 3 workers 16 days to finish a job:

[tex]16=\frac{k}{3}\\\\k=16*3\\\\k=48[/tex]

To find how long will it take 8 workers to finish the job, you must substitute the value of "k" and [tex]x=8[/tex] into [tex]y=\frac{k}{x}[/tex]. Then you get:

 [tex]y=\frac{48}{8}\\\\y=6[/tex]

Answer:

-18

Step-by-step explanation:

= |3 – 10| - (12 / 4 + 2)^2

= 7 - (3 + 2)^2

= 7 - (5)^2

= 7 - 25

= -18

Answer:

700

Step-by-step explanation:

he originally has 500 songs which is the constant. He adds 10 songs a week which is the slope. and 20 will be the number of weeks. You can make the equation y=10x+500 and replace the x with 20. 20 times 10 is 200 and 200 plus 500 is 700 which is your answer

Answer:

[tex]\large\boxed{-9y^5}[/tex]

Step-by-step explanation:

[tex]-3y\cdot3y^4=(-3\cdot3)(y\cdot y^4)\qquad\text{use}\ a^n\cdot a^m=a^{n+m}\\\\=-9y^{1+4}=-9y^5[/tex]

Answer:

Table P represents a function

Step-by-step explanation:

* Lets explain the meaning of the function

- A function is a relation between a set of inputs and a set of outputs

 in condition of each input has exactly one output

- Ex:

# The relation {(1 , 2) , (-4 , 5) , (-1 , 5)} is a function because each x in the

   order pair has only one value of y

# The relation {(1 , 2) , (1 , 5) , (3 , 7)} is not a function because there is x

   in the order pairs has two values of y (x = 1 has y = 2 and y = 5)

* Lets solve the problem

# Table P :

- In put    : 8  ,  1  ,  5

- Out put : 3  ,  7  , 4

∵ Each input has only one output

∴ Table P represents a function

# Table Q :

- Input     : 9  ,  9  ,  4

- Out put : 3  ,  5  ,  2

∵ The input 9 has two outputs 5 and 2

∴ Table Q doesn't represent a function

# Table R :

- In put    : 7  ,  8  ,  7

- Out put : 2  ,  6  ,  3

∵ The input 7 has two outputs 2 and 3

∴ Table R doesn't represent a function

# Table S :

- In put    :  1  ,  1  ,  9

- Out put :  7 ,  5  ,  2

∵ The input 1 has two outputs 7 and 5

∴ Table S doesn't represent a function

* Table P represents a function

Answer:

The answer is A I just took the test

Answer:

Piecewise function model

Step-by-step explanation:

According to the given statement a factory manufactures widgets based on customer orders. If a customer's order is for less than w widgets, the customer's cost per widget is d dollars.

customer order < widgets

If a customer's order is for at least w widgets, the customer's cost per widget is decreased by c cents for each widget ordered over w widgets. A customer's total cost is t.

customer order > widgets

We can notice that we have two different conditions. In this type of question where there are different conditions for different types of domain we use piecewise function.

With different conditions, the cost is different, therefore piecewise function model would be the best option for the situation....

namely, what is the leftover amount when the decay rate is 2% for an original amount of 75 grams after 10 years?

[tex]\bf \qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &75\\ r=rate\to 2\%\to \frac{2}{100}\dotfill &0.02\\ t=\textit{elapsed time}\dotfill &10\\ \end{cases} \\\\\\ A=75(1-0.02)^{10}\implies A=75(0.98)^{10}\implies A\approx 61.28\implies \stackrel{\textit{rounded up}}{A=61~grams}[/tex]

Answer with explanation:

The exponential decay function is written as :-

[tex]f(x)=A(1-r)^x[/tex], where f (x) is the amount of material left after x years , A is the initial amount of material and r is the rate of decay.

Given : The amount of original item remained now = 75 grams

The rate of decay = 2% = 0.02

Now, the amount of original artifact would there be left if they had not discovered it for another 10 years is given by :-

[tex]f(10)=75(1-0.02)^{10}[/tex]

Solving the above exponential equation , we get

[tex]=61.2804605166\approx61[/tex]

Hence only 61 grams original artifact would there be left if they had not discovered it for another 10 years .

Answer:

The answer is 0.4375 .

Hope this helps!