CAN SOMEONE ANSWER THIS RIGHT PLEASE!!!!!!

For the graph, what is a reasonable constraint so that the function is at least 600?

graph of y equals minus 2 times the square of x plus 40 times x plus 600

0 ≤ x ≤ 20
−10 ≤ x ≤ 30
x ≥ 0
All real numbers

They say 1 and 2 wrong I don’t know what to believe
Please answer right I need it so much

Answers

Answer 1

The reasonable constraint so the function is at least 600 would be A. 0 ≤ x ≤ 20.

How to find the reasonable constraint ?

Given the function y = -2x² + 40x + 600, we want to find the constraint for x so that the function's value is at least 600.

Since the parabola opens downward (as a is negative), the function will be at least 600 for all values of x within the interval where the parabola is above y = 600. The vertex of the parabola is at its maximum value, so we should look for the interval around the vertex (10, 800).

Looking at the given constraints, the option "0 ≤ x ≤ 20" is a reasonable constraint that ensures the function is at least 600. The parabola starts at a value of 600 at x = 0, reaches its maximum value at the vertex, and then returns to a value of 600 at x = 20.

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Related Questions

Name a point on the given line, then find the slope of the line. y + 2 = 3/2 (x + 7)

Answers

Answer:

y = (3/2)x + 17/2

Slope: 3/2

Step-by-step explanation:

y + 2 = 3/2 (x+7)

y + 2 = (3/2)x + 21/2

y = (3/2)x + 21/2 - 2

y = (3/2)x + 21/2 - 4/2

y = (3/2)x +17/2

Find the angles of the circle with the information given​

Answers

The measure of arc BC is 40⁰.

The measure of angle BPE is 50⁰.

The measure of angle BAD is 55⁰.

The measure of angle AOP is 110⁰.

The measure of angle AFB is 70⁰.

The value of arc DE is 110⁰.

What is the measure of the angles?

The measure of each angle is calculated as follows;

if arc BA = arc CD, then value of arc BC is calculated as follows;

BA + CD + BC = 180 (sum of angles of a semi circle)

arc CD = 70 (intersecting chord theorem)

70 + 70 + BC = 180

140 + BC = 180

BC = 180 - 140

BC = 40

Arc AE = 70⁰, since angle EOF = 70 (vertical opposite angles)

Arc AB = 180 - (AE + BC)

arc AB = 180 - (70 + 40)

arc AB = 70⁰

The value of angle P is calculated as follows;

arc EB = 70 + 70 = 140⁰

m∠BPE = ¹/₂ (140 - 40)

m∠BPE = 50⁰

m∠BAD = ¹/₂ (BC + CD) (intersecting chord theorem)

m∠BAD = ¹/₂ (40 + 70)

m∠BAD = 55⁰

m∠AOP =  (AB + BC) (intersecting chord theorem)

m∠AOP = (70 + 40)

m∠AOP = 110⁰

m∠AFB = (AB) (intersecting chord theorem)

m∠AFB = 70⁰

The value of arc DE is calculated as follows;

DE = 360 - (180 + CD) (sum of angles of a circle)

DE = 360 - (180 + 70)

DE = 360 - 250

DE = 110⁰

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In the diagram above, the sine of angle ACB = 3/5. What is the sine of angle ANM?

Answers

Because the sine of angle ACB = 3/5, then the sine of angle ANM will be equal to 3/5 too.

Understanding Triangle

A triangle is a shape with 3 sides. There are 3 types of triangle which are:

equilateral triangle: this type has all sides equalisosceles triangle: this type has 2 sides equalscalene triangle: this one has no sides equal

From the diagram, we can see that we have a small triangle inside a big triangle. From the principle of triangle, the angle C is equal to angle N.

Let us say angle C is 30° then the angle N will also be equal to 30*. Therefore the angle ACB = angle ANM = 3/5

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Write the factored form of the polynomial function with real coefficients, a lead coefficient of 1, and zeros of -1,-3,3,1

Answers

Answer:

To find the factored form of the polynomial function, we start by using the zeros to write out the factors of the polynomial. If the zeros are a, b, c, d, then the factors are (x-a), (x-b), (x-c), and (x-d).

In this case, the zeros are -1, -3, 3, and 1, so the factors are (x+1), (x+3), (x-3), and (x-1). Multiplying these factors together gives the polynomial function:

f(x) = (x+1)(x+3)(x-3)(x-1)

To simplify this expression, we can use the distributive property to expand the factors:

f(x) = (x+1)(x^2 - 9)(x-1)

Next, we can multiply the remaining factors using the distributive property and combining like terms:

f(x) = (x^3 - 8x - 9)(x-1)

Finally, we can use the distributive property again to expand the remaining factor:

f(x) = x^4 - 9x^2 + 8x + 9x^3 - 8x - 9

Simplifying this expression gives the factored form of the polynomial function:

f(x) = (x+1)(x+3)(x-3)(x-1) = x^4 - 9x^2 + 8x - 9x^3 - 8x - 9.

Determine if the segment lengths form a triangle. if so, would the triangle be acute, right, or obtuse

4.1, 8.2, 12.2

A.No, they do not form a triangle

Answers

Answer: D

Step-by-step explanation:

An engine does 62 J of work each cycle as it absorbs 2 times as much heat as it discharges to the environment. A) Determine the efficiency of the engine in percent. B) What quantity of heat is discharged to the environment each cycle in Joules?

Answers

Answer:

A) 150%

B) 20.67 J

Step-by-step explanation:

A) To determine the efficiency of the engine, we can use the formula:

efficiency = (work output / heat input) x 100%

In this case, the engine does 62 J of work each cycle, so the work output is 62 J. We're also given that the engine absorbs 2 times as much heat as it discharges to the environment, so we can express the heat input as 2 times the heat discharged:

heat input = 2 x heat discharged

Therefore, we can rewrite the efficiency formula as:

efficiency = (62 J / (2 x heat discharged)) x 100%

Simplifying this expression, we get:

efficiency = 31 J / heat discharged

Now we need to determine the value of heat discharged. We know that the engine absorbs 2 times as much heat as it discharges, so the net heat input is 3 times the heat discharged:

net heat input = heat input - heat discharged = 2 x heat discharged - heat discharged = heat discharged

Therefore, we can express the net heat input as:

net heat input = 3 x heat discharged

Now we can use the first law of thermodynamics, which states that the net heat input is equal to the work output plus the heat discharged:

net heat input = work output + heat discharged

Substituting the given values, we get:

3 x heat discharged = 62 J + heat discharged

Solving for heat discharged, we get:

heat discharged = 20.67 J

Now we can calculate the efficiency:

efficiency = 31 J / 20.67 J = 1.50 or 150%

Therefore, the efficiency of the engine is 150%.

B) To determine the quantity of heat discharged to the environment each cycle in Joules, we can use the value we calculated above for heat discharged:

heat discharged = 20.67 J

Therefore, the quantity of heat discharged to the environment each cycle is 20.67 J.

Hope this helps!

Please assist with this question if you can! thank you

Answers

The top is x=7 and the bottom is y=4

Which ordered pair (x,y) could not be substituted in the relation below so that the relation is functional (6,7),(-2,9),(8,8),(-4,9),(x,y) a. (9,9) b. (2,9) c (7,8) d. (8,3)

Answers

None of the answer choices would result in an x-value being associated with two different y-values, the correct answer is:

e. None of the above.

what is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.

To check whether the relation is functional, we need to make sure that each x-value is associated with only one y-value.

We can see that the pairs (6,7), (-2,9), (8,8), and (-4,9) all have different x-values, and therefore are associated with different y-values. This means that the relation is currently functional.

To determine which ordered pair could not be substituted so that the relation is functional, we need to find an x-value that is associated with two different y-values.

Let's look at each of the answer choices and see if they fit this criteria:

a. (9,9): This pair does not already appear in the relation, so it's possible that it could be added without affecting the functionality of the relation.

b. (2,9): This pair does not already appear in the relation, so it's possible that it could be added without affecting the functionality of the relation.

c. (7,8): This pair does not already appear in the relation, so it's possible that it could be added without affecting the functionality of the relation.

d. (8,3): This pair does not already appear in the relation, so it's possible that it could be added without affecting the functionality of the relation.

Since none of the answer choices would result in an x-value being associated with two different y-values, the correct answer is:

e. None of the above.

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In •A find CE if BA=7.35.

Answers

The value of CE is 14.7 units.

What if the diameter of a circle?

The diameter of a given circle is a straight line which passes through the center of the circle and divides it into two equal semi-circles.

So that in the given circle with center A, we have;

CE = BD  (diameter of a given circle)

BA = EA = CA = DA    (radius of a given circle)

CE is the diameter of the circle, and BA is the radius, so that;

CE = 2*BA

     = 2*7.35

     = 14.7

CE = 14.7

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Question 5 of 10
A circle is the collection of points in a plane that are within a certain distance
from a given point in the plane.
True
False

Answers

The collection of all points in a plane that are equally spaced from a given point, referred to as the circle's center, is referred to as a circle. The radius of the circle is the distance measured from any point on the circle to its center.

A circle is described as the collection of all points on a plane that is at the same distance, or radius, from a fixed point known as the circle's center.

Every location inside the circle is, therefore, closer to the center point than the radius, and any point outside the circle is farther away from the center point than the radius.

Therefore, a circle is made up of points that are spaced out by a particular amount, known as the radius, from the circle's center.

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USE STRUCTURE The graph of a quadratic function has a vertex (2, 0). One point on the graph is (S. 9). Explain how to find another
point on the graph.
. It can be found by graphing the points given and sketching the parabola
Another point on the graph is (-1,
that passes through them. You can move over and up from the vertex and do the same on the opposite side of the line x=

Answers

It can be found by graphing the points given and sketching the parabola. Another point on the graph is (-1, 9) that passes through them. You can move over and up from the vertex and do the same on the opposite side of the line x = -1.

How to determine the other points for the given quadratic equation?

In Mathematics and Geometry, the vertex form of a quadratic function is represented by the following mathematical equation:

f(x) = a(x - h)² + k

Where:

h and k represents the vertex of the graph.a represents the leading coefficient.

Based on the information provided, the vertex is located at point (2, 0), and this ultimately implies that the axis is represented by the vertical line x = 2 that passes through the vertex.

Additionally, the parabola would be bilaterally symmetric about the above axis, and the mirror image of the point (5, 9) would also be on this parabola.

Therefore, the mirror image would have the same y-coordinate of 9 while the x-coordinate is given by:

x = 2 - (5 -2)

x = -1.

In conclusion, the required point is (-1, 9).

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Complete Question:

The graph of a quadratic function has a vertex (2, 0). One point on the graph is (5, 9). Explain how to find another point on the graph. It can be found by graphing the points given and sketching the parabola. Another point on the graph is (-1, _) that passes through them. You can move over and up from the vertex and do the same on the opposite side of the line x=

There are three colored cookie jars. One jar is blue, another green and the last one pink. The blue jar contains 7 chocolate chip and 12 sugar cookies. The green jar contains 6 chocolate chip, 8 sugar and 5 peanut butter cookies. The pink jar contains 15 chocolate chip, 9 sugar and 13 peanut butter cookies. One of the three cookie jars is chosen at random. The probabilities that the blue jar, green jar, or pink jar will be chosen are 1⁄2 , 1⁄4 , and 1⁄4 , respectively. A cookie is then chosen at random from the chosen jar. What is the probability that the pink jar was chosen, if it is known that the cookie was a sugar cookie?

Answers

The probability that the pink can was chosen because the cookie was a sugar cookie is about 0.282, or about 28.2%.

Why do you think probability?  

Probability is simply how likely something is to happen. When we are uncertain about the outcome of an event, we can talk about the probability of certain outcomes - how likely they are. The analysis of events driven by probabilities is called statistics.  

We can use Bayes' theorem to calculate the probability that the pink can was chosen because the cookie was a sugar cookie. Let P(P) be the probability that the pink jar was chosen, P(S) be the probability that the chosen cookie  was a sugar cookie, and P(S|P) be the probability that a sugar cookie was chosen given that the pink one was chosen. Then we have:

P(P|S) = P(S|P) * P(P) / P(S)

We can calculate all these probabilities as follows:

P(S|P) = 9 / (15 9 13) = 9/37

This is the probability of choosing a sugar cookie, given that the pink can was chosen.  

P(P) = 1/4

This is the probability of choosing the pink can.

To calculate P(S), we need to consider all possible options for choosing a sugar cookie. We can choose a sugar loaf from a blue, green  or  pink jar. So:

P(S) = P(S|B) * P(B) P(S|G) * P(G) P(S|P) * P(P)

where P(S|B) is the probability of choosing a sugar cookie from the blue jar, P(G) is the probability of choosing the green jar, and so on. We can calculate all these probabilities as follows:

P(S|B) = 12 / (7 12) = 12/19

P(S|G) = 8 / (6 8 5) = 8/19

P(S|P) = 9 / (15 9 13) = 9/37

P(B) = 1/2

P(G) = 1/4

P(P) = 1/4

So,

P(S) = (12/19) * (1/2) (8/19) * (1/4) (9/37) * (1/4) = 271/444

We can now calculate P(P|S) using Bayes' theorem:

P(P|S) = (9/37) * (1/4) / (271/444) ≈ 0.282

Therefore, the probability that the pink can was chosen because the cookie was a sugar cookie is about 0.282, or about 28.2%.

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100 Points! Algebra question. State whether each function is a linear function. Explain. f(x)=4x^2. Please show as much work as possible. Only looking for an answer to question B. Photo attached. Thank you!

Answers

The function 2x+y=10 is not a linear function because it is not in the form of y=mx+b, where m and b are constants. The function f(x) = 4x² is not a linear function because it does not have a constant rate of change.

What is function?

In mathematics, a function is a rule that assigns a unique output value to each input value in a specified set. The input values are often called the independent variable, while the output values are called the dependent variable. Functions are commonly denoted using functional notation, such as f(x), where x is the input and f(x) is the output. Functions are used to describe relationships between variables, such as in algebraic expressions or equations. They can take many forms, including linear, quadratic, polynomial, exponential, trigonometric, and logarithmic functions. Functions are fundamental to many areas of mathematics, science, engineering, and economics, and have many applications in real-world situations.

Here,

A. The function 2x+y=10 is not a linear function because it is not in the form of y=mx+b, where m and b are constants. In other words, the function is not a straight line, because y is not directly proportional to x. Instead, it is a linear equation in two variables x and y, where the graph of the equation is a straight line.

B. The function f(x) = 4x² is not a linear function because it does not have a constant rate of change. In a linear function, the rate of change of the function (slope) is always constant, meaning that for any two points on the line, the change in y over the change in x is always the same. However, in this case, the rate of change of the function depends on the value of x, and thus the graph of the function is not a straight line. Instead, it is a quadratic function, where the graph is a parabola.

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Find the positive solution to each equation.

If the solution is irrational, write the solution using square root or cube root notation.
Equation
Positive Solution

Answers

The solutions are explained in the solution below.

1) t³ = 216

t = ∛216

t = ∛6×6×6

t = 6

2) a² = 15

a = √15

a = 3.87

3) m³ = 8

m = ∛8

m = ∛2×2×2

m = 2

4) c³ = 343

c = ∛343

c = ∛7×7×7

c = 7

e) f³ = 181

f = ∛181

f = 5.65

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Luka and Dana are making cookies for the Westland pantry. They only have up to 6 hours to bake! They can bake 32 chocolate chip cookies in an hour(x), and 20 snickerdoodle cookies in an hour(y). They need it make at least 150 cookies. Which inequality represents the situation?

Answers

Option 3. x+y≤6 , 32x + 20y ≥ 150. The inequality 32x + 20y ≥ 150 ensures that they bake enough cookies to meet the requirement of making at least 150 cookies.

The right disparity that addresses what is happening is:

32x + 20y ≥ 150

To see the reason why, we want to consider the quantity of treats Luka and Dana can heat in every hour and the all out number of hours they need to prepare. We should expect they have H hours to prepare, where H = 6 for this situation.

Luka and Dana can prepare 32 chocolate chip treats in 60 minutes, so they can heat a sum of 32x treats in x hours. Likewise, they can prepare a sum of 20y snickerdoodle treats in y hours. To make no less than 150 treats, they need to prepare a mix of chocolate chip and snickerdoodle treats in every hour, with the end goal that the complete number of treats they make in H hours is something like 150.

Thus, the imbalance 32x + 20y ≥ 150 guarantees that they heat an adequate number of treats to meet the necessity of making somewhere around 150 treats.

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100 Points! Algebra question, photo attached. Please show as much work as possible. Thank you!

Answers

Answer is down below!

Step-by-step explanation:

5 x2 =`5 xj2

Ian deposited $1,000 into an account that earns 5% simple annual interest. Jackson deposited $1,000 into an account that earns 5% interest compounded annually. The deposits will be made on the same day, and no additional money will be deposited or withdrawn from the accounts. Which investment earns more interest after 10 years? How much more interest?

Answers

Ian earned $500 in simple annual interest after 10 years, while Jackson earned $1,128.89 more in interest than Ian after 10 years due to compound interest.

Ian's account earns simple annual interest, which means that he earns 5% interest on his initial deposit every year. After 10 years, he will have earned

interest = principal x rate x time

interest = 1000 x 0.05 x 10

interest = $500

Jackson's account earns interest that is compounded annually, which means that he earns 5% interest on his initial deposit plus any interest that has already been earned every year. After 10 years, he will have earned

[tex]balance = principal x (1 + rate)^{time}[/tex]

balance = 1000 x (1 + 0.05)¹⁰

balance = $1,628.89

To find out how much more interest Jackson earned than Ian, we need to subtract the interest Ian earned from the balance in Jackson's account

$1,628.89 - $500 = $1,128.89

Therefore, Jackson earned $1,128.89 more in interest than Ian after 10 years.

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Find the area of the shaded region
Round to the nearest tenth

Answers

Answer:

160

Step-by-step explanation:

9 x 18 = 162

Since the last number is not 5 and is lower it rounds down to 160.

A rectangle is shown on the coordinate plane.

Answers

The coordinates of the vertices of rectangle are (-3,3),(6,3),(6,-3), (-3,-2).  the rectangle is 9 units long and 5 units wide.

Describe Graph?

In mathematics and computer science, a graph is a collection of vertices (also called nodes) and edges that connect pairs of vertices. Graphs are used to represent various types of relationships between objects, such as social networks, road networks, electrical circuits, and many others.

Vertices in a graph are typically represented by circles or points, while edges are represented by lines or arcs that connect pairs of vertices. A graph can be directed, where each edge has a direction, or undirected, where edges have no direction.

The coordinates of the vertices of rectangle are (-3,3),(6,3),(6,-3), (-3,-2).

To find the length and width of the rectangle, we need to determine the distance between the points on the x and y axes.

The length of the rectangle is the distance between points (-3,3) and (6,3) on the x-axis:

Length = 6 - (-3) = 9 units

The width of the rectangle is the distance between points (-3,3) and (-3,-2) on the y-axis:

Width = 3 - (-2) = 5 units

Therefore, the rectangle is 9 units long and 5 units wide.

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Find the interest
earned:
$9,900 for 24 months
at 7.8% annual simple
interest

Answers

To find the simple interest earned on an investment, we use the formula:

I = P * r * t

Where:

I = simple interest earned

P = principal amount invested

r = annual interest rate (as a decimal)

t = time period of investment (in years)

In this case, we have:

P = $9,900

r = 7.8% = 0.078 (since the rate is given as an annual rate, we need to divide by 100 to get the decimal equivalent)

t = 24 months / 12 months/year = 2 years

Substituting these values into the formula, we get:

I = 9,900 * 0.078 * 2

= $1,545.60

Therefore, the simple interest earned on the investment is $1,545.60.

plot 1 2/5 - 1/2 - 2 1/10 on a number line

Answers

A graph of the data set is shown in the image attached below.

What is a number line?

In Mathematics and Geometry, a number line simply refers to a type of graph with a graduated straight line which comprises both positive and negative numbers that are placed at equal intervals along its length.

This ultimately implies that, a number line primarily increases in numerical value towards the right from zero (0) and decreases in numerical value towards the left from zero (0).

In this scenario and exercise, we would use an online graphing calculator to graphically represent the given data set on a number line as shown in the image attached below.

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Simplify (m2n)3. .... that is M to the 2nd power times n in parenthesis to the 3rd power

Answers

(m^2*n)^3=m^6n^3
For more questions like this, use Symbolab, it’s free, easy to use, and shows the steps.

Help !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answers

The calculated value of the perimeter of triangle ABD is 48 units.

Calculating the perimeter of the triangle ABD

Given that

A(1, 1), B(17, 1) and D(17, 13)

To find the perimeter of triangle ABD, we need to find the lengths of its three sides and add them up.

We can use the distance formula to find the lengths of the sides.

The distance between points A and B is:

AB = √((17 - 1)^2 + (1 - 1)^2) = √(256) = 16

The distance between points B and D is:

BD = √((17 - 17)^2 + (13 - 1)^2) = √(144) = 12

The distance between points D and A is:

DA = √((1 - 17)^2 + (1 - 13)^2) = √(256 + 144) = √(400) = 20

Therefore, the perimeter of triangle ABD is:

AB + BD + DA = 16 + 12 + 20 = 48

So, the perimeter of triangle ABD is 48 units.

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Find a value of Z such that .6318 of the area lies between negative Z and Z.

Answers

The required value of z such that 0.6318 of the area lies between −z and z is 0.89.

If 0.6318 of the area lies between −z and z, then the area outside of this range is (1-0.6318) = 0.3682.

Since the normal distribution is symmetric, the area to the left of −z is equal to the area to the right of z. So, the area to the right of z is (0.3682 / 2) = 0.1841.

Using a standard normal distribution table, we can find the z-score corresponding to an area of 0.1841 to the right of z:

z = invNorm(0.1841) ≈ 0.89

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70+15=85 125+85=210 Write into an equation.​

Answers

Answer:

Brother asked a very good question.

We can write the given arithmetic expressions into equations as follows:

70 + 15 = 85

This can be written as an equation: x + 15 = 85, where x is the unknown value that needs to be determined.

125 + 85 = 210

This can be written as an equation: y + 85 = 210, where y is the unknown value that needs to be determined.

These equations can be used to solve for the unknown values x and y by performing inverse operations (such as subtraction or division) to isolate the unknown variable on one side of the equation.

The usual price of 4 pencils was m Meihua bought 36 pencils and was given a discount of n for each pencil how much did she pay altogether

Answers

The total amount that Meihua pays for the 36 pencils is  9M - 36n

How much did she pay altogether?

We know that a set of 4 pencils has a price M, then the price of each single pencil is:

P = M/4

And we also know that Meihua got a discount of n in each pencil, so the price that she pays for each pencil is:

P = M/4 - n

Now we know that she buys 36 of these, so the amount that she pays is given by the product:

amount  = 36*(M/4 - n)

amount = 9M - 36n

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PLSSS HELPP
This figure represents the base of a fish tank that is 8 inches high.

What is the volume of the fish tank?

Responses

378 in³
378 in³

408 in³
408 in³

480 in³
480 in³

768 in³

Answers

The volume of the fish tank is 408in³

What is volume?

Volume is defined as the space occupied within the boundaries of an object in three-dimensional space. It is also known as the capacity of the object.

The volume of regular object is given as;

V = base area × height

base area = 13 × 6

= 78in²

The area of the cut out section = 1/2(a+b) h

= 1/2 × (3+6) × 6

= 9× 3 = 27in²

The actual area of the base shown = 78-27

= 51 in²

therefore the volume of the tank will be

V = 51 × 8

= 408in³

therefore the volume of the tank is 408in³

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6/5/8 - 2/3/8 estimate

Answers

The estimated value of the expression [tex]6/5/8 - 2/3/8[/tex] is approximately [tex]0.5082[/tex].

How can we calculate the multiple fraction?

In Mathematics, fractions are defined as the parts of a whole. The whole can be an object or a group of objects

To estimate 6/5/8 - 2/3/8, we can convert the fractions to decimals and then perform the subtraction.

6/5/8 is equivalent to 6 ÷ 5.8, which is approximately 1.0345. 2/3/8 is equivalent to 2 ÷ 3.8, which is approximately 0.5263.

Therefore, 6/5/8 - 2/3/8 is approximately 1.0345 - 0.5263 = 0.5082.

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dertermine if T:R^3-R^2 defined as T(x1,x2,x3)=(x1+x3,2x2-x3) is a linear transformation

Answers

Yes, [tex]T(x_1,x_2,x_3)=(x_1+x_3,2x_2-x_3)[/tex] is a linear transformation.

What is a linear transformation?

A linear map is a mapping V to W between two vector spaces in mathematics, more specifically in linear algebra, that keeps the operations of vector addition and scalar multiplication. The more broader example of modules over a ring also uses the same terminology and definition are Module homomorphism.

What is homomorphism?

The maps between algebraic objects are known as homomorphisms. Group homomorphisms and ring homomorphisms are the two basic categories. In Other words ,homomorphisms of modules, algebras, and vector space homomorphisms, which are also known as linear maps.

Two properties must be satisfied  if  T is a linear transformation:

T(x+ y)  = T(x) + T(y) for any x, y in [tex]R^3.[/tex]

homogeneity is T(ax) = a T(x). For all scalars a and x in [tex]R^3.[/tex].

let a consider

L.H.S,T(x+y)= T[tex]((x_1+y_1),(x_2+y_2),(x_3+y_3[/tex]))= [tex](x_1+y_1+x_3+y_3, 2x_2+2y_2-x_3)[/tex]

R.H.S. T(x) + T(y) =[tex](x_1+x_3, 2x_2-x_3) +(y_1+y_3, 2y_2-y_3) =(x_1+y_1+x_3+y_3, 2x_2+2y_2-y_3).[/tex]

L.H.S.=R.H.S.

So,The first property is satisfied i.e. T(x+ y)  = T(x) + T(y) for any x, y in [tex]R^3.[/tex]

For homogeneity,

let a consider ,

L.H.S. T(ax) = T[tex](ax_1,ax_2,ax_3) = (ax_1+ax_3, 2ax_2-ax_3).[/tex]

R.H.S. a T(x) = a T[tex](x_1,x_2,x_3) = a(x_1+x_3, 2x_2-x_3) = (ax_1+ax_3, 2ax_2-ax_3).[/tex]

L.H.S.=R.H.S.

So, T(ax) = a T(x). for all scalars a and all vectors x in  [tex]R^3.[/tex].are satisfied

A linear transformation from [tex]R^3[/tex]to[tex]R^2[/tex]  . is therefore represented by T.

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The factorization of x2 – 6x – 72 is:

Answers

Step-by-step explanation:

x^2 -6x - 72   =  ( x-12)(x+6)

If you can't do this 'in your head'  you can always use Quadratic Formula

  with  a = 1      b= - 6    and   c= -72

      to find roots   x = 12 and  -6      then write the equation as above

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