I really need help please figure this out I need it by Monday October 17 2021 please help me thank you

The amount of **money** that June has is $40 and Maria has $400. The total amount is $440.

From the information, **John** has $400 June has 1/10 of that and June has 0.1 of Maria’s money.

Therefore the **amount** that June has will be:

= 0.1 × John's money

= 0.1 × 400

= 40

June has 0.1 of Maria’s money. Let Maria's **money** be x. Therefore, 0.1 × x = 40

0.1x = 40

Divide

x = 40 / 0.1

x = 400

Maria's money = 400

Therefore, the total **amount** that they have will be:

= Amount that John has + **Maria's** amount

= 400 + 40.

= $440

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1. The number of miles a car can travel on a gallon of gasoline is related to its fuel

efficiency. Fuel efficiency depends on many factors including the speed of the car.

Driving at very slow or very fast speeds uses more gas than driving at a moderate

speed, which means fuel efficiency can often be modeled by a quadratic function.

For a certain model of car the fuel efficiency is maximized at 32 miles per gallon

with a speed of 55 mph. At 65 mph the car can get 31 miles per gallon.

The **quadratic equation** that models the fuel efficiency of the car is given by:

y = -0.01(x - 55)² + 32.

What is the equation of a parabola given it’s vertex?The equation of a quadratic function, of **vertex (h,k),** is given by the following rule:

y = a(x - h)² + k

In which the **parameters **of the equation are described as follows:

For a certain model of car the fuel efficiency is maximized at 32 miles per gallon with a speed of 55 mph, hence the **vertex **is given by:

(h,k) = (55,32).

Hence the **rule **is:

y = a(x - 55)² + 32

When x = 65, y = 31, hence we can find the** leading coefficient** a as follows:

31 = a(65 - 55)² + 32

100a = -1

a = -0.01.

Hence the **equation **is given by:

y = -0.01(x - 55)² + 32.

What is the missing information?The problem is incomplete, hence we suppose that it asks for the quadratic model of the car's fuel efficiency.

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Halp me with this 1 question Brainliest to correct answer

**Answer:**

A is correct.

[tex]g = \frac{p}{6} - 15[/tex]

**Step-by-step explanation:**

[tex]p = 6g + 90[/tex]

[tex]6g = p - 90[/tex]

[tex]g = \frac{p}{6} - 15[/tex]

So A is correct.

an exponential function f has a 3-unit growth factor of 1.55. what is the 1-unit growth factor for f ? what is the 2-unit growth factor for f ? what is the 1 2 -unit growth factor for f ?

Given that 3 unit factor growth is 1.65

Then 1 unit growth factor = (1.65)^1/3 = 1.181

2 unit growth factor of F = 1.396

And 1/2 unit growth factor = (1.65)^0.5/3 = (1.65)^0.53/3 = (1.65)^1/6

= 1.0875

or 1/2 unit growth of factor f =** 1.0875**

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Question 1 1 pts If g and h are both positive numbers, which statement is true about the x-intercepts of the function f(x) = (x - g)(x - h). the function has no x-intercepts both x-intercepts are negative one x-intercept is negative and one x-intercept is positive both x-intercepts are positive

for the function

f(x) = (x -g)(x - h)

To get the x-intercept, we can equate the function to zero

f(x) = (x -g)(x - h) = 0

so (x - g) = 0

which gives **x = g**

And (x - h) = 0

which gives **x = h**

Since we have been told that g and h are positive, then it follows that both intercepts are positive

Handmade socks, knitted using pure cashmere wool, are very expensive to buy. Rowena buys cashmere wool in 20g balls. Each ball of cashmere wool costs her £1.42 . She pays her sister £8 to knit each pair of socks. 135g of cashmere wool is used to knit each pair of socks. Rowena sells 40 pairs of cashmere socks at £18.95 per pair. What is her percentage profit? Give your answer correct to 2 significant figures.

The **percentage profit** for the so is that were made is 40.32%.

From the information, **Rowena** buys cashmere wool in 20g balls. Each ball of cashmere wool costs her £1.42 and she pays her sister £8 to knit each pair of **socks**. The cost will be:

= (£1.42 × (20 + 135) + (40 × £8)

= £220.1 + £320

= £540.1

The selling **price** will be:

= £18.95 × 40

= £758

Therefore, the **percentage profit** will be:

= Profit / Cost price × 100

= (758 - 540.1) / 540.1 × 100

= 217.9 / 540.1 × 100

= 40.32%

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Sophie opened a picture of a tree on her cell phone screen and zoomed in 2.5 times. The new scale of the photograph became 1:168. What would be the scale if Sophie zoomed the image in 4 times?

1:105

The picture is 1 and the tree is 168 making the ratio 1:168. If you divided the 1 by the zoom which is 2.5 you get 0.4. Then you multiply 0.4 by 4 which is the new zoom. The final ratio is 1.6:168 and if you simplify, by dividing both sides by 1.6, you get 1:105

The picture is 1 and the tree is 168 making the ratio 1:168. If you divided the 1 by the zoom which is 2.5 you get 0.4. Then you multiply 0.4 by 4 which is the new zoom. The final ratio is 1.6:168 and if you simplify, by dividing both sides by 1.6, you get 1:105

Solve 5(x + 2) = 26 - 3x

x = ...

Answer: x = 8

5(x + 2) = 26 - 3x

5x + 10 = 26 - 3x

take the smallest x away from both sides.

2x + 10 = 26

take 10 off 26

2x = 16

divide 16 by 2 to get x

x = 8

5(x + 2) = 26 - 3x

5x + 10 = 26 - 3x

take the smallest x away from both sides.

2x + 10 = 26

take 10 off 26

2x = 16

divide 16 by 2 to get x

x = 8

If the water level rises 12 centimeters every 7 minutes and you record the data point of (21,y), what is the value of y? Use slope to justify your answer.

The **rate **of **change **of the given **relation **is 1.7142 the value of y is 36 **centimeters**.

The **rate **of **change **is the change of a **quantity **over 1 unit of another quantity.

Most of the **time** the **rate** of **change **is the change with **respect **to time.

As per the given **question**,

The **water level **rises 12 centimeters every 7 minutes.

0 minutes → 0 cm

7 minutes → 12 cm

14 minutes → 24 cm

21 minutes → 36 cm.

Since, (21,y) best **match **with the above **condition**.

So, 21 minutes is the **time taken** while 36 cm is the level of water.

**Slope **or rate of change = 12/7 cm/minutes.

And, 36/21 = 12/7 (proved)

Hence "The **rate **of **change **of the given **relation **is 1.7142 the value of y is 36 **centimeters"**.

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If line m is parallel to line n then determine the value of x.

If line m is **parallel to line **n then determine the value of ** x= 65°**

2x+50° = 180°

2x = 180° - 50°

2x = 130°

x = 130°÷2

**x = 65°**

**Parallel lines** are lines that do not cross or meet at any point in the plane. They are always parallel and equidistant from each other. Parallel lines are non-intersecting lines. We can also say that parallel lines meet at infinity.

**Properties of Parallel Lines**

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The **value **of x is 25. It states that when two **parallel **lines are cut by a transversal, the resulting **alternate **interior angles or alternate exterior angles are **congruent**.

**Parallel **lines are coplanar straight lines that do not intersect at any point in geometry. **Parallel** planes are planes that never meet in the same three-dimensional space. Parallel **curves **are those that do not touch or intersect and maintain a constant minimum **distance**.

**Parallels**, paralleling, and paralleled are all word forms. a **countable **noun A parallel is something that is similar to something else but exists or occurs in a different **place **or at a different **time**.

Parallel lines are lines that are always the same **distance **apart in a plane. **Parallel **lines never cross. **Perpendicular **lines are those that intersect at a right angle (90 degrees).

Two different interior **angles**.

- (2x - 10)°

- (65 - x)°

Because **lines **m and n are parallel and because of the **alternate **angles theorem.

We can see that.

2x - 10 = 65 - x

2x + x = 65 + 10

3x = 75

x = 75/3

**x = 25**

Hence, the **value **of x is 25.

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The number of bricks, B , needed to build a wall of length L feet and uniform height H feet can be found by the equation B = 7 L H . A wall of uniform height that is 16 feet long is constructed using 280 bricks. What is the height, in feet, of the wall?

The **height** **of the wall** in feet is 2.5 feet.

According to the question,

We have the following information:

The number of bricks, B , needed to build a wall of length L feet and uniform height H feet can be found by the equation B = 7 L H .

A wall** **of uniform height that is 16 feet long is constructed using 280 **bricks**.

So, we have:

B = 280 and H = 16

Putting these **values** in the given **expression**:

B = 7LH

280 = 7*L*16

L = 280/(7*16)

L = 280/112

L = 2.5 feet

Hence, the height of the wall in feet is 2.5 feet.

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What is the inverse of function f?

f(x)=10/9x+11

The inverse is 9/10(x-11)

This table shows the ratio of the number of adults to the number of students on a class field trip. A 2-column table has 4 rows. Column 1 is labeled Number of adults with entries 1, 2, 4, 6. Column 2 is labeled Number of students with entries 7, x, 28, y. Which values are missing from this table? x = y =

**Answer:**

** x = 14 and y = 42**

**Step-by-step explanation:**

From the two entries that have both the numbers od adults*A) and students(S) we find a ratio of S/A to be 7 for both cases. We'll make the assumption that is the same ratio in the two cases involving x and y.

To find x and y, multiply the number of adults by 7 to find the number of students,

** x = 14 and y = 42**

A S Ratio S/A = 7

1 7 7

2 x 14

4 28 28

6 y **42**

v/8 - 3.1 = -15.9 solve for v

v = -102.4

Hope this helped

Hope this helped

[tex]\frac{v}{8} -3.1=-15.9[/tex]

Simplify equation:

[tex]\frac{1}{8}v+-3.1=-15.9[/tex]

[tex]\frac{1}{8} v-3.1=-15.9[/tex]

Add 3.1 to both sides:

[tex]\frac{1}{8} v-3.1+3.1=-15.9+3.1\\[/tex]

[tex]\frac{1}{8} v=-12.8[/tex]

Multiply both sides by 8:

[tex]8\times(\frac{1}{8} v)=(8)\times(-12.8)\\-102.4[/tex]

[tex]\fbox{v=-102.4}[/tex]

1. Compare two spheres. The first has a diameter of 4 inches.

The second sphere has a diameter of 428 inches.

Determine the ratio of the volume of the large sphere to the volume

of the small sphere.

Ratio =

The ratio of the volume of the large sphere to the volume

of the small sphere is 1226 .

How to find the volume of 3d objects?**V = 4/3 r³ **is the equation to calculate a **sphere's volume.** The volume of a sphere is calculated using the following formula: 4/3rds of Pi and the cube of the sphere's radius. A sphere is a** geometrical object** that is a **three-dimensional **equivalent of a two-dimensional **circle** . A sphere is a collection of points in three-dimensional space that are all located at the same **r distance **from a particular point. The radius of the sphere is equal to r, and the provided point is its **center.** In the works of the ancient Greek mathematicians, spheres are first mentioned.

In several branches of mathematics, the **sphere** serves as a basic building block. Also found in nature and industry are spheres and nearly-spherical shapes.

**Volume **sphere = 4/3πr³

Volume of sphere 1

D = 4 inches

r = 2 inches

So,

Volume = 4/3π(2)³

= 33.51 inches³

Volume of sphere 2

D = 428 inches

r = 214 inches

So,

Volume = 4/3π(214)³

= 4.11 ×[tex]10^{7}[/tex] inches³

Ratio

= sphere 2: sphere 1

= 4.11 ×[tex]10^{7}[/tex] : 33.51

= 1226

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Lance the alien is 5 feet tall. His shadow is 8 feet long. At the same time of day, a tre shadow is 24 feet long. What is the height of the tree?

**Answer:**

15 ft

**Step-by-step explanation:**

5:8

?:24

8x3=24 SO 5x3=15

Members of our lacrosse team raised 1672.50 to go to a tournament they rented a bus for 1068.50 and budgeted $37.75 per player for meals write and solve an equation which can be used to determine x

The** equation is** 1068.50 + 37.75 x = 1672.50 and the value of** x is **16.

Members of our lacrosse team raised 1672.50 to go to a **tournament**.

Amount raised by lacrosse team = 1672.50

Let the number of players in lacrosse team = x

They rented a bus for 1068.50 and budgeted $37.75 per **player** for meals. So, mathematically this can be represented as

1068.50 + 37.75 x

Total amount paid by lacrosse team = 1068.50 + 37.75 x

As **amount** raised by lacrosse team and the amount paid by lacrosse team are equal so we can write

1068.50 + 37.75 x = 1672.50

37.75 x = 1672.50 - 1068.50

37.75 x = 604

x = 16

Thus, we can conclude that the value of **x is** 16 and the** equation is ** 1068.50 + 37.75 x = 1672.50.

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the following data were obtained for days required to reach 230 lb and backfat thickness of a sample of 10 pigs: pig no. days to 230 lb, days backfat thickness, in 1 164 1.0 2 180 1.1 3 158 1.2 4 160 1.4 5 198 1.2 6 172 1.3 7 186 1.1 8 178 1.3 9 178 1.4 10 186 1.0 calculate the covariance between days to 230 lb and backfat thickness.

The **covariance** value between two variables says, days to 230 lbs and backfat thickness, is 0.48..

The **covariance** value between any two variables is used to determine whether the variables are changing in the same direction. It relates the two variables if they are from different data sets.

Let X and Y be two data sets with different values. Then the covariance between X and Y is given by

**Cov ( X,Y) = ****∑**** ( ****X**** ****ᵢ****– u) ( Y****ⱼ****-v )**** ****/n**

Where are Xᵢ's elements in the X data set

Yⱼ→ elements of the Y set

μ → mean of X data set

ν→ mean of Y set

n→ total number of observations

As we have given two sets of data, let days to 230 lbs and backfat thickness be sets X and Y, respectively.

Now, we shall calculate the mean value for X and Y sets.

Mean of X set = (164+158+180 + +160+198+172+186+178+178+186)/10= 176

The mean of Y set =

(1.0+ 1.1+ 1.2+ 1.4+1.2+1.3++1.1+1.3+1.4+1.0)/10 = 1.2

Xᵢ– μ - 12 -18 4 -16 22 -4 10 2 2 10

Yⱼ– ν -.2 - .1 0 .2 0 .1 -.1 .1 .2 -.2

(Xᵢ - μ)(Yⱼ- ν) 2.4 1.8 0 - 3.2. 0 0.4 1 4.4 -2

∑ (Xᵢ - μ ) ( Yⱼ - ν) = 4.8

put all the values of variables in above formula we get,. Cov( X, Y) = 4.8 / 10 = 0.48 ..

So, the Covariance between the days of 230lb and backfat thickness is 0.48 ..

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Translate "the sum of m and 2.33 multiplied by s is 77.77" into an algebraic equation.

Although part of your question is missing, you might be referring to this complete question "**Translate** the sum of m and 2.33 multiplied by s is 77.77 into an algebraic equation. Do not solve the equation."

Translated **equation** is **m + 2.33s = 77.77**

An algebraic equation is the equality of two expressions formulated by applying a set of variables such as addition, subtraction, multiplication, division, raise to power, etc. The **phrase** " the sum of a and b" into an algebraic expression is determined as a+b. It deals with symbols and these symbols deal with each other through variants.

Hence, the equation translated is a **linear **equation of algebraic equations. **m+ 2.33s = 77.77 **

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for a between-subjects experiment, any factor that increases the variance within treatments also increases the likelihood of finding a significant difference between treatments.

The answer is FALSE. The likelihood of discovering a substantial difference between treatments does not increase with any **factor** that makes treatments more variable.

In experiments, you create situations where several treatments (such a are used in order to investigate the impact of an independent **variation**.

Every participant in a between-**subjects** or between-groups design only experiences one condition, and group differences between participants in various conditions are compared. In contrast, no participant in a within-**subjects** design is exposed to every condition.

As a result, the likelihood of discovering a substantial difference between treatments is not increased by **factors **that enhance **variation** within treatments.

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At its peak, an online shopping site was selling 420 items per second. How many items would it sell in 30 minutes

The **online shopping** site would sell 756000 items in 30 minutes

Given

The online site would sell** 420 items in a second**

To find number of items it would sell in 30 minutes first of all convert **minutes into seconds** and continue to solve the problem.

To convert minutes into seconds just **multiply minutes with 60**.

because one minute is equal to 60 seconds.

Therefore 30 minutes = 30 x 60 seconds

30 minutes = 1800 seconds

Say the number of items sold by sire in 30 minutes be x.

now according to the condition in **equation**

420 x 1800 = x

x= 756000

Hence the site will sell** 756000 items in 30 minutes.**

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Pleaseee help will give brainliest to best and most detailed answer!

f(x) = x3 + 5x2 –32x – 7 is divided by x – 4

I really need help on this, and please show me every step!

**Answer:**

[tex]f(x)=x^2+9x+4+\dfrac{9}{x-4}[/tex]

**Step-by-step explanation:**

Long Division Method of dividing polynomials

Given:

[tex]\textsf{Dividend: \quad $x^3+5x^2-32x-7$}[/tex]

[tex]\textsf{Divisor: \quad $x-4$}[/tex]

[tex]\large \begin{array}{r}x^2+9x+4\phantom{)}\\x-4{\overline{\smash{\big)}\,x^3+5x^2-32x-7\phantom{)}}}\\{-~\phantom{(}\underline{(x^3-4x^2)\phantom{-b))))))..)}}\\9x^2-32x-7\phantom{)}\\-~\phantom{(}\underline{(9x^2-36x)\phantom{)))..}}\\4x-7\phantom{)}\\-~\phantom{()}\underline{(4x-16)\phantom{}}\\9\phantom{)}\\\end{array}[/tex]

Therefore:

[tex]f(x)=x^2+9x+4+\dfrac{9}{x-4}[/tex]

Definitions

**Dividend**: The polynomial which has to be divided.

**Divisor**: The expression by which the divisor is divided.

**Quotient**: The result of the division.

**Remainder**: The part left over.

Can anyone help me with this?

hope the attachment helps you!

Yes the attachment is correct

A Construction worker needs to pour a rectangular slab 29 feet long 16 feet wide and 6 inches thick how many cubic yards of concrete does the worker need

We have a rectangular slab.

The volume of concrete is equal to the volume of this rectangular prism.

The volume of this prism is equal to the product of its length, width and thickness.

Then, we can write:

[tex]\begin{gathered} V=L\cdot W\cdot t \\ V=29ft\cdot16ft\cdot6in \\ V=29ft\cdot16ft\cdot(6in\cdot\frac{1ft}{12in}) \\ V=29\cdot16\cdot\frac{6}{12}ft^3 \\ V=232ft^3 \end{gathered}[/tex]We convert the thickness from inches to feet by multiplying it by a unit ratio 1 ft / 12 inches, which has a value of 1, as 1 ft = 12 inches, but let us change the units from inches to feet.

When we have all the measures in feet we can have the volume in cubic feet.

**Answer: 232 cubic feet.**

The point (-2, K) lies on the circle x^2 +y^2= 20. Find the values of k. Show all the steps

I will plug in the POINTS including k in the equation each point in its place either x or y.

[tex]( - 2)^{2} +( {k})^{2} = 20 \\ 4 + {k}^{2} = 20 \\ k^{2} + 4 - 20 = 0 \\ {k}^{2} - 16 = 0 \\ (k + 4)(k - 4) = 0 \\ \\ k + 4 = 0 \\ \\ or \\ \\ k - 4 = 0 \\ \\ k = 4 \\ or \\ k = - 4[/tex]

ATTACHED IS THE SOLUTION

Moores law predicts that the number of transistors that can be fit on a microchip will increase 40% every year. If microchips from a given year could hold about 922,200 transistors, how many transistors could fit on a microchip 19 years later?

If necessary, round your answer to the nearest whole number.

[tex]5.51134751094269\times10^8[/tex] **transistors** could fit on a **microchip** [tex]19[/tex] years later.

**Moore's law** predicts that the number of transistors that can be fit on a microchip will increase 40% every year.

So the expected** increment** every year [tex]40\%[/tex].

Microchips from a given year could hold about [tex]922,200[/tex] transistors.

We have to find ow many transistors could fit on a microchip [tex]19[/tex] years later.

From the question total years are [tex]19[/tex].

Since the number of transistors is increased every year so number of transistors of previous year added. So we can see that the number of transistors that can be fit on a microchip is increased **compound**. We can find the total number of transistors by multiplying number of transistors with the rate assembled by one. So we can use the formula

So the number of transistors could fit on a microchip 19 years later.

Number of transistor = Microchips from a given year could hold about transistors(1 + increment rate)^total years

Number of transistor [tex]=922,200(1+\frac{40}{100})^{19}[/tex]

Number of transistor [tex]=922,200(1+\frac{2}{5})^{19}[/tex]

Number of transistor [tex]=922,200(\frac{5+2}{5})^{19}[/tex]

Number of transistor [tex]=922,200(\frac{7}{5})^{19}[/tex]

After evaluating the expression using the calculator we get

Number of transistor [tex]=551134751.094269[/tex]

Number of transistor [tex]=5.51134751094269\times10^8[/tex]

Hence, [tex]5.51134751094269\times10^8[/tex] transistors could fit on a microchip [tex]19[/tex] years later.

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f(x) = 4x³ + 6x² – 3x − 4

g(x) = 4x - 3

Find (f - g)(x).

Solution of the given **algebraic expression** (f-g)(x) is equal to 4x³+6x²-7x-1

**Algebraic equations** are defined as mathematical statements in which two **algebraic expressions** are set equal to each other. An algebraic expression usually consists of variables, **coefficients** and constants. **Algebraic equations** can be of many types, such as monomial algebraic equations, binomial algebraic equations, quadratic algebraic equations, polynomial equations.

Given in the question that the value of f(x) is equal to 4x³+6x²-3x-4

and the value of g(x) is equal to 4x-3.

Subtracting g(x) from f(x) we get,

(f-g)(x)= 4x³+6x²-3x-4-4x+3

4x³+6x²-7x-1

Therefore, solution of the given **algebraic expression** (f-g)(x) is equal to 4x³+6x²-7x-1.

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Leah can run 720 yards in 4.5 minutes. How many yards can she run in 2 minutes at the same rate?

A. 350

B. 320

C. 160

D. 110

D. 110 or b hope it helped!

Leah can **run **320 yards in 2 minutes at the same rate as she can run 720 yards in 4.5 minutes. The correct option is B. 320.

To find out how many **yards **Leah can run in 2 minutes at the same rate, we can use the concept of **proportionality**.

Let's set up a proportion based on the information given:

720 yards / 4.5 minutes = x yards / 2 minutes

To find the value of x, we can cross-multiply:

4.5 minutes × x yards = 720 yards × 2 minutes

Now, **divide **both sides by 4.5 to solve for x:

x yards = (720 yards × 2 minutes) / 4.5 minutes

x yards = 320 yards

So, Leah can **run **320 yards in 2 minutes at the same rate.

The correct answer is B. 320.

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Johnny drew a scale drawing of an apartment. The scale he used was 8 inches : 3 feet. In the drawing, the broom closet is 16 inches long. What is the length of the actual closet?

The actual **closet **has a **length **of 6 feet

The scale of the **drawing** is given as

Scale = 8 inches : 3 feet.

Also, we have the **length **of the closet to be

Length = 16 inches

So, we have

16 inches : Actual length = 8 inches : 3 feet.

Multiply the ratio by 2

So, we have

16 inches : Actual length = 16 inches : 6 feet.

By comparison, we have

Actual length = 6 feet

Hence, the **length **of the actual **closet **is 6 feet

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WILL GIVE BRAINLY Given f(x)=3x-1 and (f∘g)(x)=x^2, find g(x)

[tex]\begin{cases} f(x)=3x-1\\\\ (f\circ g)(x) = x^2 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ (f\circ g)(x)\implies f( ~~ g(x) ~~ )~~ = ~~3( ~~ g(x) ~~ ) - 1\implies \stackrel{(f\circ g)(x)}{x^2}~~ = ~~3g(x)-1 \\\\\\ x^2+1=3g(x)\implies \cfrac{x^2+1}{3}=g(x)[/tex]

Answer:

The value of g(x) is x^2/3x-1 of a function f(x)=3x-1 and (f∘g)(x)= x^2

i think this is the answer hope this was helpful!

The value of g(x) is x^2/3x-1 of a function f(x)=3x-1 and (f∘g)(x)= x^2

i think this is the answer hope this was helpful!

Find all real numbers r for which there exists exactly one real number a such that when (x+a)(x2 +rx+1)

is expanded to yield a cubic polynomial, all of its coefficients are greater than or equal to zero.

All **real numbers **r for which there exists exactly one real number a such that when (x+a)(x² +rx+1) is expanded to yield a cubic **polynomial**, all of its **coefficients **are greater than or equal to zero are given as r = -1

A **real number **is a quantity of a continuous number that can express a distance along a line in mathematics.

Hence the workings for the above result are indicated as follows:

Expanding the brackets, we see that we want the following three **inequalities **to be true.

a ≥ 0 (constant term) (1)

ar + 1 ≥ 0 (coefficient of x) (2)

a + r ≥ 0 (coefficient of x²) (3)

If r ≥ 0, then any a ≥ 0 meets the requirement of (1), (2), and (3).

It remains to address r < 0. In this case, note that (3) immediately suggests (1)

Hence, we only need to look at (2) and (3). Given that r < 0, **inequalities **(2) and (3) are equivalent to the following.

a ≤ − 1/r (4)

a ≥ − r (5)

Therefore, we seek all values of r < 0 such that there is exactly one **real number **satisfying the **expression **−r ≤ a ≤ −1/r (6)

Thus −r = −1/r ,

This implies r = ±1.

Given that r < 0 we have r = −1.

This implies that the **only corresponding value **of a is a = 1.

It only remains to follow that (x + 1)(x² − x + 1) = x³ + 1, which has no **negative coefficients**.

Learn more about **real numbers**:

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**Full Question:**

Find all real numbers r for which there exists exactly one real number a such that when (x+a)(x² +rx+1) is expanded to yield a cubic polynomial, all of its coefficients are greater than or equal to zero.

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