what is the factorization of the polynomial below 2x^2+20x+50

The required suitable unique real numbers a, b, and C that represents the given graph are a = -1, b = 1, and c = 3.

A graph can be defined as a pictorial representation or a diagram that represents data or values.

The polynomial function of the graph which is given in the question:

f(x) = (x - a)²(x - b)(x - c)

This means that the polynomial's factors are as follows:

(x - a)², (x - b) and (x - c).

The polynomial solution will now be the values of x when the components equal zero.

This shows that a, b, and c are polynomial function solutions. This indicates that the x-intercepts are a, b, and C.

The x-intercepts on the graph are;

-1, 1, and 3.

Hence;

a = -1

b = 1

c = 3

Thus, the required suitable unique real numbers a, b, and C that represents the given graph are a = -1, b = 1, and c = 3.

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The simplest radical numbers according algebra properties are listed below:

In this question we find two cases of radical numbers that must be simplified in accordance with algebra properties, especially those related to square roots, and factor decomposition. Now we proceed to simplify each expression:

Case A: √175

Step 1 - Use factor decomposition

√175 = √(5² × 7)

Step 2 - Apply algebra properties to simplify the expression

√175 = √5² × √7

√175 = 5 × √7

√175 = 5√7

Case B: √(5 / 12)

Step 1 - Use factor decomposition

√(5 / 12) = √[5 / (3 × 2²)]

Step 2 - Apply algebra properties to simplify the expression

√(5 / 12) = √5 / √(3 × 2²)

√(5 / 12) = √5 / [√3 × √2²]

√(5 / 12) = √5 / (2√3)

Step 3 - Rationalize the resulting expression

√(5 / 12) = (√5 × √3) / 6

√(5 / 12) = √15 / 6

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Answer:

0.001.

Step-by-step explanation:

So, P (X=150) = P (X>150) - P (X>151) So, P (X>151) = (151-120)/sqrt (120) = P (Z>2.8299) = 0.5- 0.4977 = 0.0023 So, the probability of getting 150 customers in a day = 0.0033- 0.0023 which is equal to 0.001.

Answer:

80y by 40y

Step-by-step explanation:

The area of the rectangle that maximizes the enclosed area is

80 by 40

80+80=160

40+40=80

160+80=240 yards.

Answer:

Step-by-step explanation:

The fraction that Martha has left to make matching throw pillows is 3 1/6 yards.

A fraction is simply a piece of a whole. The number is represented mathematically as a quotient where the numerator and denominator are split. In a simple fraction, the numerator as well as the denominator are both integers. On the other hand, a fraction appears in the numerator or the denominator of a complex fraction.

From the information given, Martha had 12 2/3 yards of drapery material. She used 3/4 of it to make the drapes for her family room.

Since 3/4 has been used, the fraction left will be gotten by subtracting the fraction that had already been used from the total fraction that she has. In this case, it should be noted that this will be illustrated as:

Total fraction that Martha has - The fraction that has been used

= 1 - 3/4

= 1/4

Therefore, the fraction will be the fraction that is left multiplied by the yards of drapery material that Martha had at the beginning. This will be:

= Fraction unused × Total yards of material

= 1/4 × 12 2/3

= 1/4 × 38/3

= 19/6

= 3 1/6

The fraction is 3 1/6 yards.

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Answer:

∴ x = 4.5, y = 5.5

Step-by-step explanation:

x + y = 10 ------ equation (1)

x - y = -1 ------ equation (2)

Add equations (1) and (2):

(x + y) + (x - y) = 10 + (-1)

   x + y + x - y = 10 - 1

                  2x = 9

                 ∴ x = 4.5

Substitute x = 4.5 into equation (1):

4.5 + y = 10

     ∴ y = 5.5  

Answer:  The correct answer is No, you cannot determine who is winning.

Step-by-step explanation:

Why you cannot determine who is winning:

Your Score is -3

Your friend is -6 (plus or minus 4 points)

Therefore, the deviation in your friend's score is:

(-6+4) to (-6-4)

Your friend's score is between -2 to -10

Whether the highest or lowest score wins:

Since your score is -3, your friend would win with -2 or tie with -3 (or lose with less). Therefore, you cannot determine who would win based on your friend's score being in a range between -2 to -10.

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From the given details of Manuel wooden crates , where x represents the number of crates and y represents the cost then equation represents the given situation is given by y = 10x + 5.

As given in the question,

Given details of Manuel wooden crates :

Number of crates  : 2       4       6         8

Cost in dollars       : 25     45      65      85

From the given table where x represents the number of crates and y represents the cost :

(x₁ , y₁) = ( 2, 25)

(x₂ , y₂) = ( 4, 45)

For the given situation equation is given by :

( y - y₁) / (x- x₁) = (y₂ - y₁) / (x₂ - x₁)

⇒( y - 25) / (x -2) = (45 -25)/ (4 -2)

⇒(y -25) / (x -2) = 10

⇒y - 25 = 10(x-2)

⇒ y = 10x +5

Therefore, from the given details of Manuel wooden crates , where x represents the number of crates and y represents the cost then equation represents the given situation is given by y = 10x + 5.

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The derivative of the inverse function of the differentiable and increasing function f(x), g(x) = f⁻¹(x), at x = -2, g'(-2) is [tex]\dfrac{1}{7}[/tex]

An inverse function reverse the effect of the operations of a function, such that when the output of a function is the argument of its inverse function, the input of the function is obtained.

The derivative of an inverse function is found using the formula;

[tex]\dfrac{d}{dx} f^{-1}(x) = \dfrac{1}{f'[f^{-1}(x)]}[/tex]

When x = -2, we get, from the table included in the question;

g(-2) = f⁻¹(-2) = 1

Therefore;

[tex]\dfrac{d}{dx} f^{-1}(-2) = \dfrac{1}{f'[f^{-1}(-2)]} = \dfrac{1}{f'[1]}[/tex]

[tex]g'(-2) = \dfrac{d}{dx} f^{-1}(-2) = \dfrac{1}{f'[1]}[/tex]

In the included table, f'(1) = 7, from which we get;

[tex]g'(-2) = \dfrac{1}{f'[1]}=\dfrac{1}{7}[/tex]

[tex]g'(-2) = \dfrac{1}{7}[/tex]

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Answer:

x=66 degrees so b

Step-by-step explanation:

132 and 2x are corresponding angles so 132=2x and x=66 degrees

Answer:

x = 66

Step-by-step explanation:

2x = 132

Divide by 2 on both sides

x = 66

The function that models the distance in miles is f(x) = 20 + 50x

The domain of the function is 0 ≤ x ≤ 2

Information given in the question include

Ali and Kris are driving to a city that is 120 miles from their house

They have already traveled 20 miles

they are driving at a constant rate of 50 mi/h

The function can be modeled as a linear function

A linear function consists of functions where the variables has exponents of 1. The graph of linear functions is a straight line graph and the relationship is expressed in the form.

y = c + mx

Definition of variables to suit the problem to be solved

y = total mileage covered = f(x)

m = mileage covered per hour = 50 mph

x = number of hours

c = miles already covered = 20 miles

The function is substituted to the equation as below

y = c + mx

f(x) = 20 + 50x

The domain refers to the reasonable time spent during the journey

this is time at minimum distance = 0 to time at maximum distance = 120

f(x) = 20 + 50x

0 = 20 + 50x

-20 = 50x

x = -2/5

since there is no negative time, it is more reasonable to start with zero

120 = 20 + 50x

120 - 20 = 50x

50x = 100

x = 100 / 50

x = 2 hours

A reasonable domain is 0 ≤ x ≤ 2

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Part (a)

To verify a point is on the line, we plug the coordinates into that equation. If we get a true result at the end, then we've confirmed the point is on that line.

For point A(-5,0) we have x = -5 and y = 0 pair up together. Let's plug these coordinates into x - 3y = -5 to get the following steps shown below.

x - 3y = -5

-5 - 3(0) = -5

-5 - 0 = -5

-5 = -5

We get a true statement at the end, which confirms x-3y = -5 is true when x = -5 and y = 0. Therefore, point A is definitely on the line x - 3y = -5

Repeat similar steps for B(4,3) to show that this point is also on x - 3y = -5

x - 3y = -5

4 - 3(3) = -5

4 - 9 = -5

-5 = -5

This confirms point B is also on this line.

We have confirmed line AB is x-3y = -5

I'll let you check to see if C(-3,9) and D(1,-3) are on the line 3x+y = 0. The steps will follow the same outline as shown above.

=========================================================

Part (b)

The standard form equation Ax+By = C has the slope m = -A/B, where B cannot be zero.

The equation x-3y = -5 has A = 1 and B = -3 to give us a slope of m = A/B = -1/(-3) = 1/3

Meanwhile the equation 3x+y = 0 gives the slope of A/B = -3/1 = -3

Notice that slopes 1/3 and -3 multiply to -1. This is one useful property of perpendicular lines. Their slopes always multiply to -1; this is assuming neither line is vertical, nor horizontal.

Put another way, the slopes are negative reciprocals of one another. We flip the fraction and flip the sign to go from 1/3 to -3, or vice versa.

I recommend using GeoGebra to plot out the points mentioned, and forming the lines AB and CD. This helps give a quick visual confirmation the lines are perpendicular.

----------

To summarize: We found the slopes of line AB and CD to be 1/3 and -3 respectively. The slopes multiply to -1 which is sufficient to conclude the lines are perpendicular.

=========================================================

Part (c)

We have this system of equations

x - 3y = -5

3x + y = 0

To represent lines AB and CD respectively.

Let's triple everything in line CD to go from 3x+y = 0 to 9x+3y = 0

Therefore, an equivalent system is this

x - 3y = -5

9x + 3y = 0

The first equation hasn't changed. After this point, we can see the y terms add to 0 since -3y+3y = 0y = 0. Therefore, the y terms cancel which lets us solve for x.

10x = -5

x = -5/10

x = -0.5

Then we use this to find y

x-3y = -5

-0.5-3y = -5

-3y = -5+0.5

-3y = -4.5

y = -4.5/(-3)

y = 1.5

The point of intersection is (-0.5, 1.5)

Notes:

-0.5 = -1/2

1.5 = 3/2

=========================================================

Part (d)

The x coordinates of points A and B are -5 and 4 in that order.

Add them up: -5+4 = -1

Divide the result in half: -1/2 = -0.5

This is the x coordinate of the midpoint of segment AB.

Repeat for the y coordinates

Add: 0+3 = 3

Divide in half: 3/2 = 1.5

The midpoint of segment AB is (-0.5, 1.5) which is exactly the result of part (c).

You should find that the midpoint of segment CD is also (-0.5, 1.5)

I'll let you do those steps.

=========================================================

Part (e)

Quadrilateral ACBD is a kite because of the perpendicular diagonals. This was confirmed in part (b).

The sides aren't all the same length (which you can confirm with the distance formula), which means we don't have a rhombus.

Answer:

a)  See below.

b)  See below.

[tex]\textsf{c)} \quad \left(-\dfrac{1}{2},\dfrac{3}{2}\right)[/tex]

[tex]\begin{aligned}\textsf{d)} \quad \textsf{Midpoint of $\overline{AB}$}&=\left(-\dfrac{1}{2},\dfrac{3}{2}\right)\\ \textsf{Midpoint of $\overline{CD}$}&=(-1,3)\end{aligned}[/tex]

e)  Kite

Step-by-step explanation:

Given endpoints:

Given equations of the lines:

To quickly check that the given equations are correct, input the x-value of a point on the line into the given equation of the line containing that point. If the y-value corresponds with the y-value of the point, the equation is correct.

Using point A to check equation AB:

[tex]\begin{aligned}x=-5 \implies -5-3y&=-5\\-3y&=0\\y&=0\end{aligned}[/tex]

As point A is (-5, 0), the equation is correct.

Using point C to check equation CD:

[tex]\begin{aligned}x=-3 \implies 3(-3)+y&=0\\-9+y&=0\\y&=9\end{aligned}[/tex]

As point C is (-3, 9), the equation is correct.

If two lines are perpendicular, their slopes are negative reciprocals.

The negative reciprocal of a number is -1 divided by the number.

Rearrange each equation to slope-intercept form, then compare slopes.

[tex]\begin{aligned}\textsf{Line}\;AB: \quad x-3y&=-5\\-3y&=-x-5\\3y&=x+5\\y&=\dfrac{1}{3}x+\dfrac{5}{3}\end{aligned}[/tex]

Therefore, the slope of line AB is ¹/₃.

[tex]\begin{aligned}\textsf{Line}\;CD: \quad 3x + y &= 0\\y&=-3x\end{aligned}[/tex]

Therefore, the slope of line CD is -3.

[tex]\textsf{As} \;\dfrac{-1}{-3}=\dfrac{1}{3}, \textsf{ the slopes are negative reciprocals}.[/tex]

Hence, the lines are perpendicular.

To find the point of intersection of AB and CD, solve the system of equations.

[tex]\begin{cases}x - 3y = -5\\3x + y = 0\end{cases}[/tex]

Rearrange the second equation to isolate y:

[tex]\implies y=-3x[/tex]

Substitute the found expression for y into the first equation and solve for x:

[tex]\implies x-3(-3x)=-5[/tex]

[tex]\implies x+9x=-5[/tex]

[tex]\implies 10x=-5[/tex]

[tex]\implies x=-\dfrac{5}{10}[/tex]

[tex]\implies x=-\dfrac{1}{2}[/tex]

Substitute the found value of x into the expression for y and solve for y:

[tex]\implies y=-3\left(-\dfrac{1}{2}\right)[/tex]

[tex]\implies y=\dfrac{3}{2}[/tex]

Therefore, the solution to the system of equations is:

[tex]\left(-\dfrac{1}{2},\dfrac{3}{2}\right)[/tex]

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Midpoint between two points}\\\\Midpoint $=\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)$\\\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the endpoints.\\\end{minipage}}[/tex]

[tex]\begin{aligned}\textsf{Midpoint of $\overline{AB}$}&=\left(\dfrac{x_B+x_A}{2},\dfrac{y_B+y_A}{2}\right)\\&=\left(\dfrac{4+(-5)}{2},\dfrac{3+0}{2}\right)\\&=\left(-\dfrac{1}{2},\dfrac{3}{2}\right)\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{Midpoint of $\overline{CD}$}&=\left(\dfrac{x_D+x_C}{2},\dfrac{y_D+y_C}{2}\right)\\&=\left(\dfrac{1+(-3)}{2},\dfrac{-3+9}{2}\right)\\&=\left(-1,3\right)\end{aligned}[/tex]

The midpoint of line segment AB is the solution to the system of equations.

Quadrilateral ABCD is a kite:

Answer:

----------------------------------

It is assumed boats leave from the same point.

Boats travel after 2 hours:

The angle formed between the two directions:

The line between the endpoints is opposite to this angle.

Use the law of cosines and find the distance:

Answer:

77.3 miles apart (nearest tenth)

Step-by-step explanation:

Given information:

Draw a diagram using the given information (see attached).

To find how far apart the boats are, model as a triangle and find the length of the missing side by using the cosine rule.

[tex]\boxed{\begin{minipage}{6 cm}\underline{Cosine Rule} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]

From inspection of the drawn diagram:

Substitute the values into the cosine rule and solve for c:

[tex]\implies c^2=40^2+56^2-2(40)(56) \cos 106^{\circ}[/tex]

[tex]\implies c^2=1600+3136-4480 \cos 106^{\circ}[/tex]

[tex]\implies c^2=4736-4480 \cos 106^{\circ}[/tex]

[tex]\implies c=\sqrt{4736-4480 \cos 106^{\circ}}[/tex]

[tex]\implies c=77.27131004[/tex]

Therefore, the boats are 77.3 miles apart (nearest tenth) after 2 hours.

Expansion of x(3.25y - 4.86) is 3.25xy - 4.86x and expansion of

x(-1.75y + 5y - 3.86) + x is 3.25xy - 2.86x. The expressions are not equivalent as the coefficient of x is different in the expanded form.

What are equivalent expressions?

In expanded form, the expression combines all its like terms.

Two expressions are equivalent if they can be simplified to the same third expression or if one of the expressions can be written like the other.

According to the given question:

The given expressions are

1. x (3.25y - 4.86)

2. x (-1.75y + 5y - 3.86) + x

Expanding each expression

1. x (3.25y - 4.86)

= 3.25*x*y - 4.86*x

= 3.25xy - 4.86x

2. x (-1.75y + 5y - 3.86) + x

= -1.75*x*y + 5*x*y - 3.86*x + x

= 3.25xy - 2.86x

Clearly coefficient of x variable is different after expansion of expressions. Therefore, the expressions are not equivalent.

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The number of machines must be made to minimize the unit cost is 280

What is second order derivative test?

A real-valued function formed on a closed or bounded interval can have its absolute maximum and lowest values determined systematically using the second derivative test. In physics, economics, and engineering, the second derivative test can be used to solve optimization problems.

We are given a function

C(x)=0.7x²-392x+60,983.

We first find the first derivative of the function we get

C'(x) = 1.4x-392

We now equate the first derivative to zero

we get

1.4x= 392

x= 280

Now we derivate the function again we get

C"(x)= 1.4

Now this value is greater than zero hence here is the greatest minimum

Hence the company must produce 280 units to minimize the cost

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Guyton can make the 72,072 distinct arrangements of the flowers.

Given, that Guyton bought 13 plants to arrange along the border of his garden.

Now, we are asked that how many distinct arrangements he can make.

As, we know that to arrange 13 plants comprised of 5 tulips, 5 roses, and 3 daisies,

we need to apply Permutations over 13 objects among which 5 are tulips, 5 are roses and 3 are daisies.

Now, on applying the permutations, we get

13 plants can be arranged in 13! ways,

but as among them 5 are tulips, 5 are roses and 3 are daisies.

So, Guyton can arrange the flowers in 13!/(5!×5!×3!).

= (6,22,70,20,800)/(120×120×6)

= 72,072

So, the arrangement can be done in 72,072 ways.

Hence, he can make the 72,072 distinct arrangements.

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The equation of the parabola as described in the task content is; f (x) = -3 (x - 5)² + 9.

It follows from the task content that the equation of the parabola which has the same shape as f(x)=-3 x² but with vertex (5,9) in the form f (x) = a (x-h)² + k be determined.

Recall that the vertex of a quadratic equations of the given form is defined by the coordinates (h, k).

On this note, the required equation of the parabola by substitution of h and k is;

f (x) = -3 (x - 5)² + 9.

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Answer:

ima need a complete question

Step-by-step explanation:

Step-by-step explanation:

every marker marks obviously a unit of 6.

so, the values go up and down from 0 in multiples of 6.

for the blue dot we count down from 0 :

5 markers

each marker is worth 6. and we are going down into the negative realm.

so, the blue dot is

- 5×6 = -30

The temperature of the solution is -7.6°F.

Temperature simply means the degree of hotness and coldness in a body.

In this case, the temperature of a solution increases by 1.7° F each minute for 2 minutes and the solution is then placed in an ice bath, and its temperature decreases by 11°F over the next minute.

The current temperature will be:

= (1.7 × 2) - 11

= 3.4° - 11°

= 7.6°F

The temperature is -7.6°F.

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The required answer is [tex](7^{\frac{1}{3}})^{3} = (7^{\frac{1}{3}*3}) = 7^1=7[/tex].

what is a power rule for exponents ?

[tex](a^m)^n=a^{mn}[/tex] is known as the power rule for exponents. Exponent times power is how you raise a number with an exponent to a power.

A number's exponent demonstrates how many times we are multiplying a given number by itself. 3^4, for instance, indicates that we are multiplying 3 by four. 3*3*3*3 is its expanded form. The power of a number is another name for an exponent. It could be an integer, a fraction, a negative integer, or a decimal.

[tex](7^{\frac{1}{3}})^{3} = (7^{\frac{1}{3}*3}) = 7^1=7[/tex]

Here, we have used the rule [tex](a^m)^n=a^{mn}[/tex]

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Answer:

12x + 9

Step-by-step explanation:

Combine the line terms

3x + 5x + 4x = 12x

8+ 2 -1 = 9

12x + 9

The equation of the parabola from the vertex is f(x) = -3(x - 5)² + 9

The equation of the function is given as

f(x) = 3x²

Also, we have

Vertex = (5, 9)

The form of the equation is given as

f(x) = a(x - h)² + k

Where

Vertex = (h, k)

This means that

Vertex = (h, k) = (5, 9)

Substitute (h, k) = (5, 9) in the equation f(x) = a(x - h)² + k

So, we have

f(x) = a(x - 5)² + 9

This also means that

f(x) = -3(x - 5)² + 9

The -3 is gotten from f(x) = -3x²

Hence, the equation of the parabola is f(x) = -3(x - 5)² + 9

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If Jerry currently has 20 points and likes to pick up the pink sticks, earning 4 points every turn, and his cousin has a strategy to catch up to jerry from 0, by getting all the red ones, earning 6 points per turn, then their scores will be tied after 10 turns from now on, and their points then will be 60.

As per the question statement, Jerry and his cousin are playing a game where they pick up colored sticks and Jerry currently has 20 points picking up the pink sticks which earn him 4 points every turn. His cousin just lost all her points on the previous turn, and has a strategy to catch up by picking all the red ones, which can earn 6 points per turn.

Then we are required to calculate the number of turns from now on, after which, their scores will be tied, and the magnitude of their points at that instant.

Let us assume that, their scores will be tied after "x" turns from now on.

Then, [{20 + (4 * x) = (6 * x)]

Or, [(20 + 4x) = 6x]

Or, [20 = (6x - 4x)]

Or, (2x = 20)

Or, [x = (20/2)]

Or, (x = 10)

That is, their scores will be tied after 10 turns from now on, and their points then will be [(20 + 4x) = (20 + 40) = 60].

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Answer: 4

Step-by-step explanation: